That may be true for 1e and 2e. In 4e, the "fighter" can be a warlord. In 3e he can take the occasional level of rogue to get those charisma based skills.
Our own thinking is often too rigid. We could do with less asking, "Why cannot <favorite class> do <thing I want to do> and instead ask, "What classes do I need to use to build a character that can do X?"
We are more flexible when we remove the class as the basis for the character concept, and instead have a character concept that we put into practice using classes as building blocks.
Yep. This is how my players have always approached it. They usually look over the classes just to be inspired and then come up with some kind of concept they like. Then I suggest some stuff that would make that work and they take a look at the options.
Add in Pathfinder's archetypes and really anything is achievable. They were really one of the single best concepts for cutting down extreme option bloat and yet greatly expanding meaningful choices.
I also really liked the hybrid system in 4E. I don't think it worked perfectly but it made for some really neat concepts. In a Dark Sun group I did everyone was a hybrid. Monk/Psion, Rogue/Warlock, Bard/Battlemind and Ranger/Warlord. The eladrin Monk/Psion was really cool.