I want to step out of the weeds of this debate about fighter/warrior classes.
Let's begin by looking at a different, contemporary RPG:
Cthulhu Dark. Here is the character building process from that system:
Choose a name and occupation. Describe your Investigator.
Here's the reason why your character's
occupation matters:
To know how well you do at something, roll:
* One die if the task is within human capabilities.
* One die if it’s within your occupational expertise.
Imagine adapting this system from Mythos RPGing to FRPGing: if I call my warrior character a
samurai that gives my character a bonus die when riding, shooting my bow, fighting with a katana and (if the table has bought into a certain conception of the courtly life of samurai) composing a haiku. If I call my warrior a
gladiator then I get a bonus die when fighting - even with weird weapons - and when revving up the crowd and maybe (if the table has bought into a certain conception of gladiators) when leading a slave revolt.
13th Age uses
just this approach to PC backgrounds and their role in non-combat resolution; Cthulhu Dark just generalises it to the whole of the game. (And a mechanically more sophisticated approach to this sort of free descriptor-style of PC build can be found in Robin Laws's HeroWars/HeroQuest RPG.)
Now the reason original D&D needed only a single "fighting man" class was because it didn't have class abilities beyond
what weapons can you use,
what armour can you use, and
what magic can you perform. The fighter class is the one that answers
all,
all,
none.
There would be no point in an alternative, invented-today D&D having multiple fighter/warrior classes unless it was also going to have intricate and distinct lists of class abilities associated with each (along the lines of 4e D&D, or 5e D&D subclasses). But just as likely, as a design, would be to stick with a single fighter class (probably called
warrior because I think that sounds less like a boxer and has a bit more resonance) and then just use some sort of background/free descriptor mechanic to distinguish gladiators from samurai from Robin Hood-style foresters etc.