I'd rather start with classes I'd straight-up drop:
1. Monk. The flavor of this class has always been mismatched. It dates back to the late 70s and early 80s, when Kung Fu and Bruce Lee were very popular. The only other setting I'm aware of that consistently includes this class is the Dragon Quest series, which calls them "Fighters" (while calling fighters "Soldiers").
2. Sorcerer. This class only exists because 3e didn't go all in on spontaneous casting (granted, partially because the magic system was way better than the martial system). At this point the only thing it really does uniquely is flavor. That could be rolled into a Wizard subclass, including the difference in magic source (learned vs inherent).
3. Warlock. Again, a class that was created to introduce new mechanics. The flavor of the class is deep, but just because you have flavor doesn't mean you have to have unique mechanics behind them. There's no reason for this to exist anymore; it could be a subclass of Wizard.
So that leaves us with nine.
Then it gets a bit difficult.
1. Ranger vs Barbarian. These two classes, realistically, share very, very similar design space. It overlaps so heavily, and Ranger's role as a pure hunter, woodsman, and tamer of the wilds is so close to politically incorrect compared to the "noble savage" Barbarian or Norseman that I'm inclined to keep the Barbarian, but I tend to favor game history. Quite honestly, they could be the same class just with different subclasses: One that grants spells and exploration/hunting, and the other that grants rage and durability. However, Ranger is more iconic to D&D, so I would favor that.
So we're at 8. I would prefer to stop here. I think Fighter, Rogue, Cleric, and Wizard are and always will be the core 4 classes. I think their existence is immutable at this point. That leaves us with:
2. Bard vs Druid vs Paladin. These are all hybrid classes. Bard is a Fighter/Rogue/Wizard with a music them. Druid is a Fighter/Cleric/Wizard with shapeshifting and animal theme, and Paladin is a Fighter/Cleric.
I think holy knights are iconic to the fantasy genre. Most fantasy settings have a holy knight class of some sort. I think Paladins, partially due to their alignment controversies, have become perhaps the fifth most iconic class in the game. I don't think they can be dropped.
Druid's ability set is somewhat unique, even if it is a blend. Adding natural deities not related to civilization adds a lot to the game. I also think that wild shape brings something unique to the game.
Bard, unfortunately, has to take the hit here to get to 7. Even though music does bring a lot, I think it's overall less valuable than what Druid, Paladin, and Ranger bring, though I think it's close.
That said, I think it's really close between Bard and Druid. I'd rank the classes in terms of importance to the game -- with ties -- like this:
1. Cleric, Fighter, Rogue, Wizard.
2. Paladin, Ranger.
3. Barbarian, Bard, Druid.
4. Monk, Warlock.
5. Artifacer, Assassin, Illusionist, Sorcerer.
6. Psionicist, Swordmage, Warlord.
7. Everything else.