I've been thinking, and I think the split is mainly between two groups of players.
Group 1 wants a minimum amount of RNG and a high level of player skill and character skill. Quality over quantity, only good stages should be allowed. This means that it's necessary to limit the amount of stages in order to have a good stage selection. Having 10 counter-pick stages with something like 5 bans would be a mess, so limiting the amount of stages to 9-12 seems like a good solution. This solution has several advantages, such as keeping the wildly used starter/counter-pick system used by all Smash games, Slap City, and Rivals of Aether (and potentially other games as well). There are some disadvantages, however: One is that it might not be obvious which 9-12 stages should be legal. There might be more potentially legal stages available, yet this system cannot deal with too many stages. We're left with two options: Cut down to a permanent stage list, or run with seasons. Another alternative is to have a large stage list (like 15-20 stages or so) and have both players pick 5 stages each (or something like that) in order to make up the stage list (there are other alternatives as well). I'm personally leaning towards just having a permanent stage list, though finding one will be a challenge for the community. Some people will lose their favorite stages, unfortunately.
Group 2 values stage diversity higher than group 1, which necessitates alternative systems. People have suggested random, veto, and a variety of other alternatives. People in this group seem to believe (in general) that having a huge stage list is good and that variety in stages is more important than balance and fairness in stage selection (many seem to believe that people should play multiple characters in order to off-set the issues of poor stage balancing).
I think Group 2 sacrifices too much in order to attain "variety" and a large stage list. Neither are all that important for the health of the competitive game, and the costs are too great. People have been trying to come up with several alternative systems, all with significant flaws that make them inferior to stage striking/regular counter-picking. Having lots of stages may be marginally more fun for spectators, but the benefits don't seem to be worth the cost.
If I were to make an estimation, I'd say that a majority of tournament players belong to Group 1 in some form, while Group 2 consists of a loud minority of Smashboards users and Redditors. People in Group 2 are, of course, free to try to convince people in Group 1 (and people on the fence) that their solution is the optimal one.
Because to some degree it already is random. Whoever wins the RPS gets an advantage. Sure, stage bans minimise that to some degree, but the point remains.
It's unfortunate, but the fact that both players get an equal amount of strikes tends to lead to a fairly neutral stage being picked for game 1. Random stage select could lead to things like facing a Rosalina on Town & City, or playing as Little Mac on Smashville, and such. With RPS, you'd still be able to ban the worst stages.
I agree with random stage select, but character should be chosen after stage (especially with the new UI). Otherwise the RNG actually would be really unfair, since it could just screw over your character without giving you a chance to account for the chosen stage.
Actually, it would be better for solo main players to pick their character before the stage. If you have random stage select
and pick character before the stage, then solo main players risk being hurt by the system more (if their opponent isn't a solo main player). If both pick character before the stage, then RNG is about equally likely to hurt either payer. If you think it would be really unfair if you picked character first, then maybe we shouldn't do random at all? I don't think people should feel pressured to pick up multiple characters in order to deal with random stage select.
If you want character diversity, you should (ironically enough) encourage solo-maining, I think, or at least not heavily dissuade it. It seems to me that people who play multiple characters will, in general, play multiple top tier characters, not multiple mid tier or high tier characters. When a mid tier player picks up a secondary, chances are it'll be a top tier character. I haven't researched this though and I'm basing it on what I have noticed, so I admit that I could be wrong about this.