That's what pre-made groups are for. I'm not suggesting to replace the existing LFG tool. I'm saying to implement a system along side that which would rank or score players based on some combination of completion, times, disconnects, lack of completion, etc.
People leaving before time? Their rank gets dinged. People rage-quitting? Their rating drops. This is, in many ways, what happens in arena ranking. You want your rank to go up, you stay until you finish or accept the penalty for quitting.
What this would do is add accountability, which is something that's EXTREMELY lacking under the current system. There's no risk outside of having to spend the time to find another group.
The upside of this is exactly what you said in your first sentence: You don't have control over who you group with. This evens the playing field and eliminates the rampant elitism and over-emphasis on the meta to the exclusion of all else. This is also something the M+ system BADLY needs.
And most of all, if you don't want to deal with the matchmaker at all, you could always form your own team without any of the rules or limits of your matchmaking rank, exactly how you can with any other piece of content. Any sort of ranked matchmaking would be in addition to the current setup. It would not detract from it or hurt it in any meaningful way that I can think of off the top of my head.
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Add a simple slider to the matchmaking to set the lower limit of your group. Or just form your own group. If you don't want to deal with matching, and set your own very specific criteria, then just don't use matchmaking and take advantage of LFG. There's no downside.
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It would need to be based on some kind of initial set of factors. Your first 5 or 10 runs in a season to get an initial ranking. Combined with successful completion, timers, deaths, wipes, etc. All of that measured against the iLVL you had for the run to give context of difficulty.
It seems like this should be possible. All these sorts of stats are the kinds of things that blizzard would want for their stupid e-sports push anyway. And Blizzard has more access to and accuracy for this kind of data than any addon like Raider.IO