Having multiple battle styles is a good idea. It keeps things interesting, and different styles can allow for different strategies and let different Pokémon thrive.
What is notable is that all the battle styles other than singles are optional (as far as I'm aware). Even double battles will be skipped or become single battles if you only have one Pokémon. This too is a good thing, as you don't want to be assaulted by multiple ways of playing the game as you're playing.
Alternate battle styles are relatively simple to implement, because moves and abilities and whatnot still work the same. You may need to tweak the valid targets for some moves, or add in another mechanic (e.g. rotating battlers), but ultimately they're just variations on a theme.
Triple battles are quite cluttered - there's a lot going on in them, and having so much to do slows the battle down. There's also nothing about them to make them more interesting than "like doubles but even more so". Double battles get a pass for being the first alternate battle style. Battle Royals have the same number of battlers as double battles, which is fine, and it's also interesting because the number of Pokémon with/against you are uneven. Horde battles are more of this, but with a different flavour (the opponents are all against you and are generally weaker), plus it came before Battle Royals. I've never has a rotation battle, but they seem pretty odd, and could well end up being full of "wasn't very effective" which is boring.
Inverse battles and sky battles are quite lazy ideas, really. I'm not keen on them just because they make such simple modifications to damage calculation/allowed Pokémon lists respectively. I'd call them clauses rather than battle styles, since you could apply them (even both at once) to another battle style. Also, sky battles made the developers depict all flying Pokémon as flying, even ones like Salamence and Xatu which clearly look better when standing, which is another point against it. If the games had actually treated these clauses as clauses, and allowed mix-n-matches, that would have been better.
I'd like to see multi-battler styles fleshed out some more, e.g. with more combination moves. Pokémon could have a Synch stat to go alongside Attack/Speed/etc., which affects the power of combination moves they are a part of.
There's a fine line between suggesting improvements to alternate battle styles and just wanting to change how all battles work. For example, I considered mentioning Pokémon-based PP (rather than move-based PP) and Active Time Battles (faster Pokémon can attack more frequently, rather than "I move you move" tennis matches) before realising that they're kinda off-topic here.
A new battle style could be a series of 1v1 battles, with no switching. The winner is whoever wins the majority of those match-ups. I'm not sure it would be a great style, because it would prefer raw strength and resilience and remove any tactics involving switching, but it's a suggestion.
Thinking about it, good suggestions for alternate battle styles would focus on making something different, rather than restrict what you can do. All the existing battle styles do this (except the sky battle clause). I can't really think of any alternate battle styles that I'd be interested in (I'm ignoring clauses because I consider them to be off-topic).
Also
@Hematite are triple battles in essentials?
Nope, and probably never will be. It's an amazing clutter and couldn't look good, plus it'd involve a major rewrite of a lot of code. I've said above that I don't find triple battles interesting, so it'd be a lot of work for (what I consider to be) no gain.