#1 is totally, completely, without restriction the way to go.
Personally I'd say run it off of an effects-based system and just have everyone choose their non-combat schticks separate from their combat schticks, but DnD needs its classes and #1 is the way to do that.
#2 isn't DnD. It's something that I think is cool but it's not DnD.
#3 I think goes with #1.
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Here's my take on the DMF:
1. It's not a badass normal. It's a Dumb Melee Fighter. Their major schtick in combat is that they hit shit and then that shit dies. That includes dragons, which are way too fucking big to do that to as a badass normal. Previously I had been conflating the DMF with the badass normal. This was a mistake. They are separate things.
2. Giving the DMF appropriate toys to play with is easy, so long as they follow the sort of thing that Frank pointed out earlier: they're self-sufficient at some point, they get people to do their accounting, and they live a comfortable life. They hit shit, it dies. Past that you can give them all kinds of fire-and-forget abilities that run in the background.
The Badass Normal is much more troublesome: they will lean heavily, probably exclusively, on outlandish circumstance or applications of skill. Retconning would probably be a central feature of the class. A high-level Badass Normal can parry, frustrate and generally stand toe-to-toe with the god of war because he's just that badass. Parry parry parry parry parry RIPOSTE trip. Now, he's still normal and so unless your gods are Olympian-style he can't actually kill them, but he can act as action-denial sufficient to let Brofag the Angel Barbarian put the Demon-splitter Axe through someone important's skull.
You would not give the Badass Normal to players that can't be imaginative. These people need to be able to whip out mystical tinfoil from under their hats in order to block mind rays, they need to have a vial of ultraGlue to stick the spider to its web, and yes, that means that they'll probably have to also be a gadgeteer of some kind because you can't punch a dragon to death as a Normal (even a badass normal). You can, however, put an arrow into something's eye (and have that mean something more than d8 + magic damage).
A shitty player will cause that to degenerate into slapstick. Then again, Brofag hitting EVERYTHING with his Demon-splitter Axe would do the same, but its a little more expected.
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"The Fighter should be better at fighting than everyone else"
What does that even mean? He's the best killer? What about assassinating? Or against iron golems? Or demons immune to non-holy effects?
What I'm trying to get at is that the DMF I was conceptualizing has, say, total immunity to everything at his HD-4. Period. They cost him (n HP) per minute or something, and it's the kind of cost where he can take on an army of them all day, Roland-style.
But just saying "they fight" isn't sufficient. It needs more detail.
Re: In DnD everyone mows down hordes
See above. A squad of wizards with wands is going to force you to roll that 1 eventually, or the 20s will wear you down, or whatever. There's a way to make something that the DMF can do.