Personally, I like a mix.
I loved Battlefield: Barrens as an event, but don't think it is something to do for random gear. Indeed, I liked it for progresing the story and impacting the world - something WoW needs more of (while the story of raids impacts the world in terms of lore, it doesn't really often do anything to the ACTUAL non-instanced world other than occasionally adding a thing to click on to rewatch cinematics.)
So, what's my mix?
I like the idea of gearing up through progression, but I don't think it should be the only or brutal way. On the other hand, I don't like past raid tiers just offering free loot for people to get there (as a non-raider who only gets to see LFR, it's still abhorent to me to just hand people tier loot for some reason...)
...I also don't want people to be forced to grind LFR or stuff just to get caught up.
So my preference would be for a multi-pronged method. Let people run tiers, but that in addition to the same %chance of loot dropping from bosses, they also have a 100% chance to drop a token that can be redeemed for N-1 tier's LFR. So were this in practice now...
...you'd run, say, Sha, ToT, etc., to get drops that you could trade for the LFR (non-tier) gear of the previous patch. This would give you enough gear you could augment a little and step foot into Flex of current tier. With the Stat Squish, it would be possible to run Normal SoO in ToT LFR gear. It'd be enough to get you in the door.
Likewise for LFR doing the same basic thing - though LFR should be easy and mostly for people to experience story (which is why they're getting rid of the tier models/item set bonuses. The first I don't mind, the second is kinda sad - I was one of those people that built dungeon sets in previous games, and that's mostly the way I've viewed LFR armor.)
I dunno the best way, but something that encourages doing stuff in order AND gives loot you can actually use would be nice. But you shouldn't have to grind Kara for two months before being able to jump into Sunwell. Going through the fights once and having SSC/TK level gear so you could at least step into Sunwell and contribute would be the way to do it, imo.
LFR catch-up is easier - just extend the "most kill final boss of previous wing" to raids so that people go through them all, and have the N-1 tier raid just have a 100% (or otherwise highly increased) drop rate for LFR raiders to slap enough gear together to meet the ilevel check for the current tier's LFR. Since the tuning isn't so tight, they don't need a full gear set for LFR, just enough to meet the ilevel que requirements.
Given, I also like the idea of solo content, professions, etc. as methods for players to gear up outside of raids, I LOVE the idea of dungeon sets from way back in Vanilla and BC, and I like the idea of solo and class quests, powerful drops from rares, reputations, and so on and so forth.
As long as you see stuff and get a little effort to gear ratio - but that it doesn't take months of grinding old content to get current - I'll be happy. I will say skipping to current tier is kind of weird, but some alternatives are worse than that...
Generally, if you have to grind so much that, by the time you get caught up, a new tier is out...that's probably a bad system.