There are other types of player skill that come up, however. One thing that used to be an issue in my old D&D games was "puzzles", where I'd put the characters in some sort of situation where there was a puzzle to be solved, intending for the players to solve it. This was inspired by things like my love of text adventures, and it mostly served to highlight how some players understood the way I think about puzzles much better than other players. But it does highlight the player/character divide. Having a puzzle that's resolved by a simple dice roll isn't very interesting, but having it be done at the player level can lead to characters acting "out of character" because their player is better at this sort of puzzle than they should be, or players feeling like they can't participate in this part of the game because it'd be breaking character (which is probably worse).
While I don't do this the way I used to in tabletops so much these days, these sorts of things do come up in LARPs, which are sometimes more intentional about focusing on player skill. If you've got a sufficiently-consistent genre of LARP, you have have players getting better at the mechanic over time be part of the plan; traditional SIK games are very focused on player skill in general, with the assumption that the front line will get better at Nerf guns and tactics and such and the hackers will get better at geometric constructions and the poker-based decking mechanic over time. (I assume campaign LARPs have similar characteristics.) Other times, this can end up looking more like "everyone goes to Halftime to get their cryptic crossword clues solved" or "everyone's sick of sudoku", when mechanics are too skewed in terms of player skill/enjoyment and/or too one-off for people to have a chance to improve. (This was one of several problems with my brief phase of trying to bring board game mechanics into LARPs.)
In things like a Diplomacy-style war game, having different levels of skill or understanding of the mechanic can cause big problems for immersion. A player messing up their first turn of the wargame because it's a new mechanic they didn't understand is a lot different from a seasoned general apparently backstabbing someone right out of the gate.
In some ways, player skill in non-combat areas of a game isn't as visible, but it's important to keep in mind in terms of keeping game balanced and fun and making sure that you're accomplishing what you're trying to accomplish. If your goal is to give people interesting in-character choices and having that drive the plot, a mechanic that some people are going to have trouble understanding or using effectively can make it hard for them to make the right choices for their character. Similarly, a mechanic that's frustrating or annoying for some portion of your players might lead to dropped plots for player-level reasons. Both of these are important to keep in mind to avoid putting in "cool mechanics" that don't do what you want for your game.