Alright, got time for a proper response now since there's a lot to cover. It's mostly because I think you're a solid poster why that kind of gotcha is a bit blase (though I do it too).
From an out of story perspective, the orcs, the Horde and the Warchief have been inseparable for all moments besides the backstory, so before I go into the story itself it helps to recall that they're a product and the martial society run by a chieftain is what they've been from the beginning, there's a lot of investment into it, regardless of its broader implications - while I'll defend pretty much every aspect of the orcs going forward in-setting, they are obviously not a society to aspire to but what they are is emblematic for a certain sort of experience - to change it may be better for the inhabitants in some kind of longform narrative, but much like Night Elf society would be improved if they adopted indoor plumbing it'd detract from the associations that make them appealing to the fanbase. What's good for the group in story and what's enjoyable for the audience are very different things. BFA is a peak example of this - in-story Anduin making everyone shake hands and love each other is better for everyone, but out of story it's a miserable slog that leaves everyone on every faction unsatisfied. With that disclaimer, onto the actual substantive points:
What separates the Blood Oath and the general orcish loyalty to the chieftain, which exists across all clans and is based around his fitness to lead, and people becoming Anduin adherents who mouth platitudes about love and peace is that one is an in-story element and the other is an out of story contrivance. The orcs are compelled in story on the basis of cultural practice to be loyal to their kin and to their leader and to only challenge him if they themselves would be more fit, but they also have other cultural values, spiritualism, communal spirit etc. that clash with it. Because of this you can have a character like Nazgrim who's split between the obligation placed upon him by the Oath and his own values and ultimately goes down with the ship because he prioritizes one value over another. The extremely hamhanded but conceptually fine story with Geya'rah is the same - loyalty to a successful leader is the highest virtue and her revelation involves tempering that with placing a social contract like expectation on the leader to care for the group interest. This is not a story you can ever tell with Anduin or Baine or whoever because their allegiance to the leader is not an in-story value that they have to weight agains their own values but is an element imposed upon them as a writing shorthand. Orcs follow the Warchief because of the Blood Oath and they can also choose to oppose the Warchief because of other traits, but Anduinists can never oppose Anduin because the reason they follow him in the unquestioning fashion they do is a Doylist element.I won't go through every iteration of the title, but it completely robbed the orcs of any identity beyond "Me orc, me smash what Warchief say smash", absolutely no better than "I'm a disciple of Saint Anduin with no mind of my own". Under that system, the orc is nothing but a tool for the Warchief, not even an individual. Through most of WoW's life time, you didn't roll a Warsong, or Bleeding Hollow, or anything but Generic Orc. The culture of strong orcs leading isn't enhanced in the Horde, it's grossly exaggerated almost to the point of parody. We're told the orcs are all strong and proud, incredibly resistant to any form of servitude, yet they accept having their identities and agency completely stripped from them for whatever Warchief ShitForBrains says today?
The Horde as a union of clans predated the Legion, but as I alluded to earlier, the Warchief isn't a cultural alien element to the orcs. It is identical to the preexisting relationship between the orcs and their chieftain, where the chieftain is the most capable in whatever the given clan places the most value in - fighting, obviously, but in the case of the Shadowmoon also things like spiritual guidance and with the Bleeding Hollow there's a prophetic and even hereditary element attached. Gul'dan specifically set it up because having the Shadow Council running things openly would be rejected by the orcs even when they were doped up. What gets rid of Blackhand also isn't a coup but existing cultural practices that he feels bound to - Mak'gora predates the Warchief system and it's that which Orgrim uses to unseat him, relying on extant orcish cultural norms to co-opt Gul'dan's subversion. This follows through in the other things Gul'dan can't get over - his physical weakness and sneakiness isn't appreciated by the orcs and the orcish veneration of the dead is a taboo that he is scared of breaking, hence why he opts for human bodies when it comes to the Death Knights. Far from orcish norms being trampled by the Legion, even when the orcs are at their lowest point they're still such a major element that the Legion's proxy has his system taken out from under him due to them and then has his actions tied up as a result. Hence Orgrim.The Horde is a recent aberration in Orc history, when for the majority of the race's existence they were tribes. Sure, they united to take down threats, but that doesn't lead to "Hey, let's throw everything away to become a hivemind!" Had they naturally become a Horde like entity of their own choosing, where they still had individual and clan voices, I wouldn't have a word to say.
We won't agree on this, but it gives me an excuse to talk about orcs which for all of people's bitching and Sadfang cinematics have become peripheral to both discussion in the fandom and the story itself.Since you mention Ogrim, I KNOW we'll disagree on him. Oh poor little me, forced to follow Blackhand instead of challenging him. Oh poor little me, forced to genocide the neighbors instead of stopping the Horde twisting the orcs. Oh poor little me, I'm warchief and forced to continue the bloody rampage. Oh poor little me, I'm vision questing to understand why the Azeroth natives hate us while my people are locked up.
Given the described ethos appears to be influenced by bushido, this is a massive failure. In bushido, what matters is acting with absolute commitment. If you die trying, it's every bit as respected as success. Orgrim simply goes along with everything, only once really standing up by challenging Blackhand, then immediately subsiding back into "I'm a victim of circumstances, there's nothing I can do to achieve what I think is right".
On Orgrim himself, most of the traits you pin on him aren't actually there. Orgrim isn't the one to get sad about having to wage war, Durotan is, nor does Orgrim bemoan that he has to do fighting. He doesn't bemoan his lack of action earlier - borne out of ingrained loyalty to his chieftain and ultimately solved by an orcish custom as noted because it lead to a war or the natives died. He does because it's left the orcs as nothing but a conquering army - their home is gone and their shamanism is the one custom that is ended by the Legion. He can't go back to Draenor and he can't bring his army of kill-happy grunts to plowshares, but what he can do is make sure the war is waged for their sake and not for the sake of the Legion. Consideration for whether the humans like it or not isn't something that causes him angst or that he laments doing it - he likes fighting, he also likes what he's fighting for - reintroducing shamanism, setting up a state on Azeroth and essentially everything that Thrall goes onto do, except Orgrim's version would have been on bigger territory and from what we gather more a riff on what the Blackrocks already had going than on human systems. Unlike Blackhand Orgrim is also an actual political actor - he wants the territory, he doesn't want to kill every human, he wouldn't particular care if he had to do it, mind, but he's perfectly willing to share land and cooperate with trolls and goblins. You're simultaneously giving him too little and too much credit - he did follow his values, just a bit late and he fought to the end to see them through, ultimately passing them to Thrall, but his dream was never of peaceful coexistence and it's only in his later life that he ever regretted the methods he used, hence his embrace of Thrall. Thrall had a clean slate and an untainted populace to work with without Orgrim's cultural baggage or circumstances, but their goals were the same.
This ties into the other topic which is worth its own topic. There's an easy misconception to be made between the orcs serving the Legion or Sylvanas and Orgrim and Garrosh. They're all assholes who want to take over the world after all and re: the Alliance it's not like any of these guys (except Garrosh) is any more concerned with any rules of engagement. This comes mostly because while it's extremely consistent and a fairly interesting portrayal it's also not very sympathetic from the modern perspective. What follows is a longer diatribe, but skip the following paragraph and go to the last one for the tl;dr:
Strength, both physical and of will comes from their origin as clans that varied between scavengers in caves beneath the massively more powerful Gorian Empire and the fact that everything on Draenor is able to kill them. This pushes two of their values - the communal spirit, hence loyalty to the clan and to the Chieftain, but also that everyone should be able to pull their own weight. The non-Frostwolf clans threw unfit children off the edge and Gul'dan got exiled because if you weren't able to pull your weight you'd be bringing others down and in societies that were, outside the Blackrock pretty rudimentary, that means individual ability, both natural and honed through practice. The lowest thing is being weak or unable to contribute and to do it for someone other than your community, as that's what being a slave is. Where the orcs differ from most fantasy takes on this trope is that they too are slavers and they do it for the exact same reason - on top of that, civilians pre-their arrival to Azeroth aren't really a thing. Any orc must be able to pull their weight and fight after all, so groups of people who can't do this aren't a core category worthy of protection or key to sustaining a society, but dead weight. This comes through mostly in how peons are treated, with only Thrall giving them any credit and with how the only clan to respect artisanal performance and value it in a leader is the Blackrock, incidentally also the most industrious clan and the most powerful one in every timeline on account of this. Battle prowess is good because you need to be able to do it to survive. Clan loyalty and spiritualism is good because it brings you together with others and capable people together are the only way to make it on Draenor. The orcs keeping up slavery thus isn't some kind of 'lol lore' situation but a natural extrapolation of this - to reduce someone to a slave is the lowest state you can reduce them to, far lower than death, which can be either glorious (in battle vs a stronger enemy for the good of your clan) or necessary (if of old age or frail,to remove weakness from your clan). Enslaving an enemy means you don't consider them worthy on any level.
The briefest version though is the most selfish one - one is done for the benefit of the orcs, the other is done for the benefit of someone else. The orcish values of martial prowess, glory in battle, community benefit etc. are present in one and absent in the other. One is under their control, the other isn't. Fel is a poison that whether you want it or not destroys all around you and can't coexist with a stable existence and that turns even the weakest, most useless person into a deadly threat like how we see emaciated warlocks ice fully grown ogre warriors. You can't 'really' control it or reduce its side gigs and for the orcs, as consisting of former slaves who were fairly low on their world's food chain, mastery over something and personal prowess are major virtues. A big bomb isn't going to kill your enemy less dead and something like the Heart of Y'shaarj isn't going to corrupt people any less, but the way it does so and even its production is something that's dependant on your power. This is not a nice distinction, it doesn't make the orcs better people to us as viewers, but it's a throughline that's present in every incarnation and gels completely with their origins and trajectory as a race.
Leaving this for last since it's the one there's least to say about it - it's a fairly obvious riff on the pop-culture version of the Mongol Horde. From the name down to the visual aesthetic, just replace horses with wolves and Genghis Khan with people of better moral character, lower bodycount and far fewer kids.We could also go over how "army and vestigial state" is absolutely absurd given the logistics involved, but Lord knows Blizzard hasn't the faintest clue how a military actually works.