@Cinebar
I don't see how aspirations only require the skills you mentioned.
Base game aspirations & skills needed (that aren't writing or painting):
Musical Genius: instrument skills (PIano, Violin, Guitar.
Public Enemy: Mischief, Programming, Handiness
Chief of Mischief: Mischief Skill
Successful Lineage: requires a skill of choice high enough to unlock the Mentoring option
Master Chef: cooking and gourmet cooking skill
Master Mixologist: cooking and mixology skill
Renaissance Sim: logic. Also: "Achieve level 8 in 6 skills "
Nerd Brain: logic, handiness
Computer Whiz: programming, video gaming, charisma
Freelance Botanist: gardening
Angling Ace: fishing
Joke Star: comedy, guitar, violin, piano, charisma
Friend of the World: charisma
So, if you're playing with the goal of maxing out achievements or reaching the top level of a career, there's still, in my opinion, plenty to do with a Sim's skills. And as @simgirl1010 said, sometimes it's fun to level up a skill just because. Like, say, a Sim you see as a nerd leveling up their logic because they love playing chess. Or a Sim maxing out their gardening skill because they like doing so. Or playing the guitar just because.