I have been reading this thread for a few pages and obviously most arguments that have been said many times are being recycled left right and center. Having said that, heres a short compilation of why requiring high ilevel is a good idea:
1. It quite significantly increases the CHANCE of having a good player who knows what he is doing and does good dps
2. While Ilevel is imperfect (because of sub-optimal stat allocation, set bonusses and trinkets) is is the BEST metric for skill that doesn't require a lot of time looking up. What other metrics would you use to determine if someone should be in your group or not, besides achievements (which is a whole other can of worms that people hate even more).
Having said that, heres an argument in favor of using ilevel that I havn't heard before: it helps to attract scarce tanks and healers.
Right now, with the massive overflow of dps compared to tanks and healers, it is really hard to find a group as a DPS, hence they are the ones mostly making the groups.
Even if you are a dps and require 820 (which is a totally reasonable ilevel for mythics) you will quite often find that tanks and healers will avoid your group in favor of other groups, especially since leaders are quite often willing to bargain with their requirements if it means getting a tank earlier (for me, I can require 840+ and being on the same armor profisciency as me easily without taking an excessive amount of time to find someone, whereas I am willing to accept 820+ tanks no questions asked).
To the people saying that mythics can be done at 800 ilevel: you are absolutely right!
I was one of the people who leveled in 3 days and then went mythics at ~810 item level with my guild/friends, we went in pretty much blind and cleared all mythics in 2 days by pretty much winging it without knowing the tactics, across all mythics there are only 1-2 dps checks and healer mana is rarely an issue so if you just make sure to stay out of the shit you are able to kill most bosses on the first try without to many issues.
However, those people like me are now all 845+ ilevel and I promise you that the average ~800 ilevel player right now is unable to do the feats we did, simply because they are less skilled at the game than me and my friends.
Having said that, currently gearing up is a joke:
I hit level 110 on my monk on sunday and cleared all 8 mythics on monday after spending ~15k gold on gear to get me to 805, I was a healer and leader of the group and by just telling the tank to take it slow on the pulls and keep an eye on my mana I went from 780-828 in a single day while only having a handfull of wipes across the entire instance. This was done almost fully through pugs (though I did have 1-2 guildy dps step in).The ease of doing mythics, combined with a lot of 820/835 boe pieces being sold which were obtained through world quests (before they were nerfed) makes it really easy to get 840+, which is ready to raid in heroics, with very little effort.
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Please link me the armory of your 854 character, I call massive BS on this since the highest possible Ilevel atm is 856, which is assuming you got a legendary and full 850 gear, including 3 850 slots in your artifact