I agree it would not break bounded accuracy, but I also think it adds needless complexity to the game, more stuff you have to keep track of.I mean they didn't have to keep them all the same--though if +1 hit bonus is too much it's a bit strange that Archery fighting style gives a +2, for example. But "Bounded Accuracy" has always been a misnomer--it's neither all that bounded, nor specifically about accuracy. It's more like "HP-as-Defense," where AC values scale more slowly and HP and damage values are the central focus of progression. (Or the frankly enormous number of ways to add some random or fixed-duration bonus, like spells, Bardic Inspiration, certain Paladin CD effects, the Diviner's Portent, Monk ki powers, a Battlemaster maneuver...sure, none of them are quite as reliable as "constant +1" but honestly the sheer number of them tells me a few weapons being a touch more accurate really would not break things as you imply, and almost all of them will usually provide a lot more than just +1.)
I like the way most of the weapons are set up in 5E, but if we have to change it, going to a system where all weapons are exactly the same and your type of weapon is for flavor would be better than adding more complexity IMO.
The concept of hit points and weapon damage is pretty far out to start with. Most people are going to die if they get stabbed by a dagger and someone swinging a huge axe at you is not apreciably more likely to kill you in one shot, so I don't really see a problem with changing that.