If you can suggest a better UI, by all means, our dev team would love to hear about it. But keep in mind it needs to fit on a reasonably-sized screen, and ideally have a decent fallback on mobile, too.
The way I can do it on Steam is very convenient and simple, so something like that would be nice:
I can see a full list of all my games without pagination and loading. A scrollbar makes it no problem if the screen is too small. The list is just the game titles, only text, so it takes very little space. I can then right click a game in that list on the left, and immediately choose what category to put it in. I can even select multiple games with shift-click or control-click and move them all at once. Additionally, Steam can show me which items have not yet been categorized, so it's easy to find out what I've missed. Maybe this can be implemented by expanding the current filtering functionality that's in the Itch PC client.
You can't do right clicks or control clicks on a touch screen, but you could use checkboxes and long presses to work around this.
However, a mobile fallback would not be needed to be implemented urgently, games can still be added to collections right now, although it takes more effort for the user.