For me the "classes feel like *classes*" bit has to do with some siloing. If I want to be all sneaky-sneaky infiltrate and get the goods and get out, and there's a class called "thief," then that should be the one *and the only one* to pick for that. College of Secrets, or Trickery Domain, or probably some of the warlock stuff I'm not thinking of breaks that feeling for me.
It's probably also (for me) a function of D&D ditching a lot of what made classes in the mid- to upper-levels unique: followers and demesnes. Thieves end up building guilds, fighters have castles, barbarians have hordes,…