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Access to boxes everywhere is something I'm a little torn on. I like the ease of access if I want to hatch eggs or mess with my team, but I also like the sense of place and lived-in purpose designated box areas have. I think PLA was the best example of it with a little ranch in the hub town that kept all your Pokemon so you could actually see the boxed pokemon just kind of frolicking when you came to town to change them out. It creates the sense that pokemon actually need to be handled in some way rather than just digitally plucked from the ether wherever and whenever you want.
HMs were a good concept, just terribly executed. PLA probably did the best version of it by having pretty much any pokemon interact with the world in the ways HMs would allow them. Any pokemon can cut a tiny tree or smash a rock with some kind of move and ride pokemon covered the other stuff like fly, surf, and more specialized moves.
I don't miss having to pay currency to relearn moves, especially in these new environments where I might be using the same high level pokemon for multiple different types of content. I won't want the same moveset for a tera raid as I would for a singles battle and I wouldn't want the same for a doubles battle. Especially if you just wanted to try out new movesets, having to get to a certain point in the game and have a cache of a specialized currency just to change moves back was always a pain. Same as a fucking name rater. I should be able to nickname my pokemon whenever I want.