It would change the balance of the game away from what was intended.
Thus far, dex to damage seems deliberately locked behind the rogue class.
Even if the option does become available to other characters, I would expect it come at the cost of at least one class feat. Possibly requiring multi-classing into rogue and spending another class feat to get that feature (which isn't currently an option).
I personally wouldn't do it, since I think widespread dex to damage is a plague that gradually makes strength defunct. I loathe it.
However, I must concede that +stat to damage is a significantly smaller fraction of your damage in this edition, with weapon specialization and striking making up a big chunk of your damage. As such, universal dex to damage is not a game breaker.
If you did do it, perhaps let thief rogues gain a portion of their sneak attack damage with more regularity. +precision damage on every non-sneak attack?
Or you could give them a whole new ability unrelated to damage. Maybe a cool new debuff mechanic?