Remathilis
Legend
Easier in theory than practice. Ok. My greatsword breaks. I grab the mace off the floor. I recalculate my attack (PB + Str) and damage (1d8 + STR). But wait, I also now add in the +2 damage from dueling, but I no longer can use great weapon master to get my PB to damage. Further, I no longer use the graze property, I have sap and I have to read what that does. While I'm doing all that, I missed what else has happened in the round.You land a crit. At the end of the turn, you have to switch weapons.
While other players take their turns, you have a couple minutes to study the statblock and adjust to hit bonus if necessary (would it even be necessary? You'll have the same attack bonus with most weapons, no?)
But also yeah, I'm all for making all weapons deal the same damage and representing the difference through reach, special qualities and whatnot.
It is also realistic — a dagger is as deadly as a sword, the difference is in how easy (or hard) it is to threaten your opponent without getting skewered yourself.
Not insurmountable, but your going to get a lot more "wait, I forgot to add my..." To the mix. More than there is already.