I like initiative. I like that it simulates chaos on the battlefield. I liked it more when it wasn't tied to an ability score. I also hate the idea of "counting down". I find it unintuitive regardless of nostalgia people seem to have for high initiative being better. Here's an alternative:
There are five ranks of initiative:
Extremely fast PCs roll 1d4
Fast PCs roll 1d6
Average PCs roll 1d8
Slow PCs roll 1d10
Extremely slow PCs roll 1d12
Monsters don't have a fucking die to roll. they instead just have a number from 1 to 10, rating them from fastest to slowest in combat. That's their initiative. PCs roll, monsters don't. Generally some sort of magic effect is required to roll 1d4 or 1d12 for initiative. Primarily I have fighter types roll 1d6, wizards roll 1d10, and everybody else roll 1d8.
GM counts up from 1 every round and when your number hits, call out your actions (I move here and attack this guy!). or you lose your initiative and act when you finally figure out what you want to do. This makes combat simple, fast, frenetic, it gives spellcasters more time (usually) to figure out what they want to do, and keeps people more invested in the action.
EDIT: Further musings: have armor be DR, but not static (d4 = no armor, etc). Leave weapons as they are, where bigger and heavier generally mean more damage. Roll all three dice. The middle result is initiative. If you're a fast class with a light weapon and light armor, you will go before, generally speaking, slower classes that wield larger weapons and heavier armor. If you either have heavier armor OR a bigger weapon, it won't hinder you as much as having both, and having both won't hinder as much for a faster, more combat ready class.
This also brings me to a different idea, which was presumably pioneered in Bastionland: no to-hit roll. Enemies will always be damaged, unless you give up your action for something else or are out of range. In this regard weapons and armor cancel each other out, and self-balance. You have armor and a weapon and hit points. No to hit crap and no armor class.
