Thanks for the suggestion about the front-screen picker. It only worked for the main account though, but I was lucky enough that only one of my kids needed to change skins outside of the first 6.
I'm also on latest Switch firmware, Minecraft 1.9.0, and the Confirm button is disabled in the in-game picker (apart from the first 6 skins). I'm logged into Microsoft. (When I'm not logged in, the skins doesn't even show up, just a progress indicator that never finishes, no information about why).
I'm pretty sure my Switch Minecraft slows down a bit for each world created. I'm now forcing my kids to remove old worlds (we're at 60 now, and even just starting the game takes a minute). I'll try to remove unused resource packs, and try to "remove cached packs" (never heard of it). I'll report back.