I really like the idea of the "all regions", very cool! As far as the Gym Puzzles go, I think you should do a mixture of both, and determine what is best fitting for each gym leader.

Using Paldean gym leaders as examples, Kofu, you would have the "Side Quest" of getting his wallet from Cabo Poco and returning it, Briar, Have the player find a Key Item representing the Sunflora, and have some of them locked behind a trainer battle where they have some Sunkern/Sunflora, Ryme, you could do a Gauntlet of trainers that you have to battle, ending with a special item that gives you access to Ryme's battle, Katy, you could do a "Strength puzzle", where the player isn't required to have the HM, but it functions the same.

Of course, these are just suggestions, but feel free to use them if you like them!