If you're like me, you think that the Dexterity stat is overused in 5E D&D. You've already heard/read the discussions: blah blah most important stat, blee bloo attacks damage armor class, blah blah. You're not interested in flogging that dead horse; you're interested in doing something about it.
So you've decided to start small, something that will have a comparatively low impact in the game: Initiative.
This is a great post, and I like the openness to different ideas. While I agree with the premise (DEX is over-used and too important), I am not sure initiative is the place most in need of correction.
Given that DEX is combining agility, coordination, reflexes, and balance already, this seems an acceptable place for initiative. The biggest problem, in my view, is that Dex adds to damage of DEX-based attcks. I don't mind STR to hit and damage; but I think DEX attacks should also have their damage adjustment based on STR. That's where I would start. (Admittedly, this is a theoretical idea, not one I have tried out.).