If we don't have a choice, I'm more in the "equal" balance camp. Single, Doubles, and whatever other battling style they have each have their own merits and strategies. Focusing just on Singles & Doubles, they're probably the "easiest" two to switch between so, while I see the game still mostly being Singles, there should be plenty more Doubles than there are. An easy way to explain away why there would be sudden switching it just saying that on this Route/in this Town or City that either Singles or Doubles is the main way of battling, of course then also having one or two trainers who decides to go against the norm and doing another kind of style (which doesn't need to be Single or Double). That way it keeps the player on their toes, not to get used to one battling style as the next location may switch it up, as well as providing that experience. Singles is the traditional battle method obviously, BUT competitive has mostly focus on Doubles and for the game not to enforce a Doubles mindset is creating a big cap between casual/main story play and competitive/post game.
As for how this would work for important battles like with Rivals, Gym Leaders, Villain Team, and the League, well it would all vary. For Rivals its probably the easiest, if not in a context where there's an official league battling happening than the Rivals can easily go "hey, let's make this a Double Battle". Now, if they're kind, they can give you a yes-no option, but honestly that wouldn't really be in the nature of wanting to keep the player ready for anything; a jerk/quicky rival would most certainly not give the choice. Gym Leaders are tricky as it has to be one or the other, to the point maybe splitting the Gym Leaders between all available style wouldn't be a bad idea (I remember there being one BW or B2W2 rom hack where they have 2 Gym Leaders for each of the main styles in those games: Singles, Doubles, Triples, & Rotation; singe Sword & Shield only has Singles & Doubles that makes it easier with 4 each (heck, Raihan is already a Double Battle so just need 3 more)). Villain Team, I'd imagine the grunts being Single as they normally don't have that many Pokemon to begin with while the Admins being Doubles (or Tag Team, the few times they could have those with the player also having a partner); meanwhile the Boss they could have switch between styles if you fight them multiple times, though I feel for the final battle it would be Single for dramatic effect. League is the big question mark here as, unlike the Gym Leaders, they really can't split them without feeling off (the Gym Leaders you have time between to adjust, but the Elite Four who you battle back-to-back might be a bit much at least on a normal difficulty); much like with the villain Admins & Boss, I'd say the Elite Four are Doubles while the Champion in Single (for those who like to have the last major story battle be them & their Starter against the Champion & their ace).
Obviously the more battle styles there are the more complicated this gets. Doubles is rarely seen but do happen, but any third and beyond is usually relegated as a gimmick in some location. Triple/Rotation happened in very specific spots, Sky Battles you had to seek out the trainer who was hiding someplace (and only battles once thus when all done no more Sky Battles not even in Battle Facility or multi-player), and Inverse Battles was just one trainer randomly placed somewhere like any other NPC (at least in XY, in ORAS they gave Inver a "facility" in Mauville City which was nice though also meant he was away from the post game's main location: the Battle Resort). Surely if they expand upon the presence of Double then it would mean these additional battle styles also deserve additional locations, no (going with my suggestion, wouldn't it mean there should be at least one route where they battle with this rare style)? As you see, it can quickly become a slippery slope.
I don't see why it can't simply be an option at the start of the game. Just ask the player if they want to play in single or double mode, and just have slight team edits for major fights in double mode.
I too have thought of that idea (though I personally prefer the ability to be able to switch back to an easier setting if one setting makes a battle too hard; the battle already exists in the game so why lock it off & GF usually let's the player decide their own difficulty and the option to back off of it (though one could argue that's more out of laziness than intended QoL design)).
Though, as
Hugin pointed out, while maybe you can get away with the NPC and villain grunts not being so tactical with their Pokemon made for Single Battle having no synergy is suddenly placed in a Double Battle setting, you can't get away with it for Rivals, Gym Leaders, villain Admins & Bosses, and the League. Either GF would have to design their teams so that they could operate efficiently enough in both Singles & Doubles OR would have to design a different team if Singles or Doubles. Once again, maybe they could make it so the early Rival battles, early Gym Leaders, and villain admins use the same "efficient enough" team while later Rival battles, later Gym Leaders, villain boss, and League all have two teams.
And even still, I would argue a Double Battles mode shouldn't be a standard mode, you would have to go into the options, pick it, and confirm it lest a kid may accidentally activate it and now suddenly a boss is walling them with their two Pokemon.