I think fixed damage would be kind of an unfun and uninteresting thing. It might speed things up, but it would be less exciting.
But I would be in favor of generally nixing the weapons table and instead simply making it so that a Fighter or a Rogue or a Cleric or a Wizard rolled the damage damage die in melee combat regardless of what the character's preferred choice of weapon is or if they chose to use improvised weapons or whatever.
Obviously there would need to be some variation to account for the difference between one-handed and shield or two handed or reach weapon.... But beyond that, the weapons table that favors certain weapons gets in the way more than helping. If you think it is cooler to have your character fight with a clawed glove or a spear rather than a longsword, then a weapons table that blatantly favors the longsword hampers enjoyment. I am sure a skilled fighter with any of those weapons can do roughly the same amount of "hit points" of damage across the 10-second round. Attacks need to be seen less as single swings as only getting in 1 swing every 10 seconds is pretty slow-paced combat anyway.
I might also be in favor of using the attack roll to determine how much damage is done instead of a second die roll-- but that system would just slow combat down. But it does seem odd that your attack can just barely hit someone and then do maximum damage or it can exceed their AC by 10 points and then only do 1 damage. But a system that fixes that would be over-complicated and likely involve charts and slow everything down.
