Most of those examples you used go both ways. So does it matter?
yes it does, since non-mirror matchups will be in 100%, since maps will not be symmetrical for the sake of non-mirror balance anyway.
The first reason is making best of 3/5/whatever tournaments much easier to run: in CoH when you play an odd number of games, one person only gets to play their preferred team once in the best of 3 or twice in the best of 5. You can do things like "whoever has the most VPs total going into the third game chooses their side" or something like that but that's an annoying consideration for players to deal with and especially when one side is better for the map they're playing in the last round or better because of Relic's latest patch, it sucks to be stuck with the side you don't want. Mirror matches allow competitive players in tournaments to play the side they want to play in each map, or even just in the final odd matchup (you could still have the first matches be Axis vs. Allies if you really wanted).
First of all, i don't think VP leader faction choice is as bad as you think it is (at least thats what i deduct from you bashing it all the time :-P). It's not optimal, but it's the best way imho. Also: you are saying you could even keep the first two games axis vs allies... so why include mirror matchups in the first place then? i've said numerous times: np to have mirrors in basic matches (that includes tourneys), i just would not want them in automatch. in tourneys you can choose pretty much everything, so idc... if someone chooses to have his tournament played only on maps like lyon (2p), st mere dumont, best and carpiquet.... who am i to say he's not allowed to do so?
edit: also, and i mentioned that before, depending on the map, it might not be enough to have mirrors in, since then the two players might argue over starting positions...
The second reason mirror matches are important for competitive play is that they let people get really good, like REALLY good, at the game, in less time. This opens up the competitive scene and lets a lot more players in - it lets in people who don't have enough time to become master class with both teams but who do have enough time to pick one team and learn it well.
you are probably referring to ppl becoming better faster since they only have to play one faction. i disagree with that statement already, since they are not becoming "better" faster, you are just toning down your requirements for considering a player "good".
apart from that, how is it any different from CoH? imagine someone not having enough time to play both Wehr and US to level 20, so he focused on Wehrmacht... he gets to level 20 and is content sitting there. Do you really think if he starts playing US he starts from scratch learning the faction? with all the knowledge he has from being the best with wehr, he pretty much knows all there is to know about US, and he only has to practise mechanics/hotkeys to become US level 20 as well.