No, the simple answer is that GWF represents specialized training, so specialized in fact that taking it eschews all others. Much like any of the other styles, likewise, they are specialized so that they cannot do more than one at once.
Well, there's one set of goalposts. I don't see that it does.
Fighter A has no damage on a miss ability. Every time he attacks, if he rolls below the target's AC, his attack is completely unproductive. No matter what his Str and attack bonus are, no matter what the characteristics of the target, he can make a million such attacks and never do anything.
Conversely, fighter B has GWF. Every time he attacks and rolls below the target's AC, he deals damage. If he exceeds the AC, he deals no extra damage and nothing special happens, only if he fails does anything happen because of this ability. The amount of damage he deals never varies, and has nothing to do with his weapon (unlike most every other example of damage by a melee attack). Even if the target's AC is huge, even if he rolls a 1, he never, ever fails to deal damage. He can make a million such attacks, but can never, ever spend six seconds without damaging the target. He never slips, gets dust in his eye, drops his weapon, has his blow parried or deflected harmlessly. Never.
I cannot see what kind of training could lead to this outcome.
As I said, I can see damage on a miss just fine as part of a paradigm in which that's how misses are defined, but this isn't that.