(Sorry, I thought that would read as a Humpty Dance reference.)
What I'm suggesting is: what if all weapons had the same damage dice...like maybe d6 or a d8. So whether you are using a dagger or a greatsword, you deal (d6 + ability modifier) damage with it. The difference between the two would be the dagger having the Finesse, Light, and Thrown properties, and the greatsword having the Heavy and Two-Handed properties.
The two weapons are distinct because of their properties, not because of their damage amounts. Because as others have mentioned elsewhere in these forums: hit points are supposed to be abstract anyway, and 'damage' isn't necessarily the size of the cut a weapon makes on your torso. So damage could be d6 for everything (and d8 if you're using two hands), and the issue that some people have about the rapier being 'too good' would be solved.
At the end of the day, though, the difference between a d6 and a d8 is less than 1 point (on average). So this feels like a tempest in a teapot to me anyway.