I'm all for attacks being slightly different depending on what weapon you're using (different/better crits, dealing with armor, knocking prone, whatever). I'd just also be okay if the fighter could choose to be a weapon master and get all of 'em.
If you go with something like this idea, we have a far more interesting approach. A fighter is automatically specialized in whatever he is holding. The power of this is limited by the fact you can only wield one (maybe 2) weapons at once, rather than limiting by a decision made last month.
Instead, I think "Weapon Specialization" should be a class feature that lets fighters (and only fighters) get some kind of advantage unique to any given weapon type they pick up.
While I don't disagree, this is not describe a "specialist." This would take a generalist class, and give it even more general toys.
Clerics, for example, do not gain every domain in their deity's portfolio. They have to pick two, and they can only prepare bonus spells from one of them per spell level. If they could get access to every domain in the game, they would blow the power balance scale out of whack with granted powers, and every cleric in the game would be identical.
Wizards do not get to specialize in every school of magic in the game, either. They pick only one, and that choice lets them skill bonuses and extra spells for that one school. If they got these bonuses for every school of magic, the way you suggest a fighter should get these bonuses for every weapon he picks up, then he isn't really a specialist at all...he is a textbook wizard with even more power balance issues, and every wizard in the realm would be identical.
I don't have a problem with fighters being able to do things with weapons that other characters cannot. Like you said, fighters can only use one, maybe two weapons at a time at most. I didn't think of that, and it is a very good point. I would support it in the new edition. But it does not address "specialization" of any sort...it is still a generalist approach to weapon use. At the end of the day, we are just giving more toys to the Fighter class. All fighters in the realm will be identical.
That's fine for some, I guess...it's been that way since the game was first published. I would just like options to make fighters more distinctive. I would like to find a way to make a fighter's choice of weapon matter more
to the character, not just to the situation at hand.