Just got a random set of ideas, that could potentially work with the above:
- When collecting a pair - trigger match-3 kind of game, similar to games like Call of Antia, but instead of characters you collect skills cards. Failing the game (probably HP and number of turns) will "fail" the pair collection with the monster respawning somewhere else on the map. Works with the original lore, quite well, since Kumoko did have a lot of skills, but obviously was not using all of them at once. This will also require certain affinities and careful balancing.
- You can re-assign skills any time you want, but you will also have an option to try to strengthen them using a bubble-sorting game (the one with the bottles or whatever). Works with lore, since there was Parallel Minds skill, but from what I remember Kumoko did have a limitation of how many clones she could have, and what they were able to do. I think the order in which the bubbles are completed determine the bonus applied to skills, with +10%, +5%, 0%, -5% and -10% applied depending on order. Yes there is a demerit to using this feature, too.
Both of the above imply use of 5 affinities, but I think the story had more "elements", so not sure here. It is possible that the number would be bigger or smaller, and that would affect the common vulnerable to/strong against pattern.
It is also possible, that some other similar "puzzle" games can be incorporated, too. Like, dunno, nesting or rather moving of nests could require that mini-game where you unscrew things. Nests could work as respawn areas (which matches the lore, too).