Yes, though I would personally subdivide these into "classes that have a core theme, that can be expressed many ways" vs "classes that do not have a core theme, or fail to actually support the core theme they're supposed to have."
Fighter fits into the former group. There are lots of ways to be a warrior of grit and thews, but ultimately they're all going to be doing similar things. Wizard is in the latter group. It either straight-up doesn't have a core theme ("problem-solver" is NOT a core theme, sorry!), or it does but does absolutely nothing to actually support it (academician unlocking magical secrets through research and development.)
Which is where most classes should be, albeit without too many limits on scope. As folks around here have, at times, been so fond of saying: limitations breed creativity. Of course, the better version of that phrase is that good limitations breed creativity, because not all limitations are created equal. A well-designed class has good limits on its scope, which spur on creativity in gameplay expression, rather than curtailing it.
Ooh, now here we have the spicy take. Because it seems to me that this group is purely arbitrary, merely a function of each individual player's sentiment.
That's one of the (several) reasons why I'm pretty opposed to class reductionism. I don't find that reductionism actually provides that much in the way of benefit--particularly when D&D is so ridiculously profligate with other areas of its ruleset, like the spell list--but it quite clearly has serious costs.
To trot out another cliche phrase, people like to say "less is more," but what they mean is "if you can get equivalent results with fewer tools, use fewer tools." You have to give up less than you're getting. Otherwise, less is in fact less.
100% absolutely yes. The problem is, as I've argued elsewhere, when you do this, you're going to find that D&D is already short about 5 new classes, and could need as many as eleven (with a very generously-cast net). Probably somewhere between 5 and 9, depending who specifically you ask.
E.g. three shoe-in class concepts that have strong thematic expression but which have, to one degree or another, been called out quite frequently as not getting enough via subclasses: Warlord, Swordmage, and Psion. We've all seen more than a few people call for each of those, I'm sure.