I've had some new insights for discussion.
Basically, in lieu of initiative, I think what is needed are rules (I use that term loosely) that govern a few different things.
I'm not going to address "who goes first" as one of theses, since I think a simple contest (using your task resolution system) between two characters who are trying to beat each other to initiate an action is all that is needed when such a situation arises where it matters.
Instead, I'm stepping back a bit and looking at what an action scene looks like, and how to manage the flow of that scene.
A good example of a scene I remember that comes to mind is the scene at the end of The Mummy 2 (I think) where the protagonists are in a lost temple in a jungle and are fighting a variety of opponents and facing other dangerous obstacles. Most of the characters are in different rooms from most of the other ones, but the action is all taking place at the same time. As the audience, we see the scene split up between views of each protagonist facing their challenges for a little while, and then on to another one. They aren't cycled through in a set fashion. They simply jump from one to the next based on what makes the scene flow better. You see each protagonist dealing with their own challenges more than once, because they don't linger long enough for them to totally overcome it.
This is exactly what a scene flow rules system should enable.
The things I think it needs to govern are:
A) How to split up into groups. We need to know how to split a party and the challenges they face up into units that we can switch our focus between.
B) How to know when to switch between units (or how long to stay with a particular unit), and which units to switch to. We need satisfying rules (again, I use the term loosely) so there isn't a need to devote brainpower to unguided decisions every single time you switch. It should be, if not obvious, at least intuitive.
C) We need to know how to handle interaction between units. If a character finished off their enemy, and moves in to help a friend who is being overwhelmed, we need framework that makes that easy to adjudicate.
For now, those are the ones that stand out to me.
Thoughts?