The characters are suppose to be heroes, and with the skill system what it is, why not let players use their best attributes in what is a very abstract part of the game? Why should Dexterity dominate initiative if other approaches can be justified? A Barbarian giving an intimidating roar causing the enemy to hesitate? why not?
Personally, I don't like the lack of variation.
As long as they are the same level, the party will all have the same proficiency bonus. And by a fairly low level, all characters will have the same number (+4 or +5) for their highest ability modifier. So everyone ends up having the exact same modifier to the initiative roll.
Functionally, letting the players pick a skill is the same as letting them just pick their preferred ability modifier, which is effectively the same as just rolling a d20. Every character is exactly the same. Which is fair, but not interesting. An option with greater variability will increase strategy, character specialization, group dynamics, etc. YMMV.