This is, on the surface, a kind of compelling idea (and fans of GURPS and some other games are going "Well, DUH!"). It adds another dimension to weapon selection so characters stop gravitating to the same sets of weapons. Heck, it could even inspire play: when your fighter is faces with an oozy opponent who ignores his slashing attack, he might be forced to grab an improvised weapon to finish the fight. That's cool, dramatic and thematic! A total win!
The reality is that the fighter is just going to carry around 3 weapons, one of each type. The opportunity cost of doing so is fairly low (encumbrance? for a fighter? I laugh!) and the payoff is high enough to allow it. Or if the payoff isn't high enough, then it hardly matters, does it?
This is one of those unfortunate design traps that I like to call Golf Bag Tactics. The idea actually has its roots in D&D, back in earlier editions when the vulnerabilities of different creatures were sufficiently wide and varied that a common solution was to carry an array of weapons. Even if you didn't count magic items, a well equipped fighter had his normal sword, a backup sword, a silver sword, a cold iron sword and a non-metal sword, and that was just for starters. It meant the fighter could choose just the right weapon for the fight, which theoretically felt clever and tactical. Unfortunately, all it really felt like was a golf bag full of swords. That idea of the vulnerabilities as drivers of RP and excitement existed, but never really materialized in the face of this.
All of which is to say, be careful of anything that looks like it adds interesting tactics and decisions during a fight which can be trivially short-circuited by choices outside of the fight. Otherwise, you might be one left holding the bag.
1 - Or he might carry some multi-purpose weapons, like an axe with a backspike. How the system handles mixed damage - like blunted edges or stabbing vs slashing with a sword - invites many options.
2 - Part of this was also the fault of a TERRIBLE understanding of the role of dramatic weaknesses in adventure design.