I'm actually a little unclear now. So far I've only seen responses to the initiative idea, but none on the alternative dice mechanic, and I feel the need to understand that mechanic a little better before passing judgment as, say, "hashed" or "not hashed".
For example, how would one get a bigger die? Is it another stat that determines it? Or do you just have to choose? Taking the time to choose which die to roll would probably distract from the fictional choices you have to make. If no other stat determined which die to roll, you could simply start high and let it get bigger every time you got a hit, and smaller every time you got a miss. So say you start with a d8, then you rolled a 4, your next die would be a d6 (meaning that your +x stat would help you succeed much more next time), but if you rolled a 5, then your next die would be a d10 (meaning you'd be somewhat more likely to get into trouble). Any +2 or +3 stat would be extremely powerful at lower die levels.
It seems to me that there's some real value to your game play right there, even without any initiative element thrown in. I suppose if you find yourself unsure about how the PvP situation should play out, who should go first, and you can't seem to find the right questions that As If was talking about in order to find the most logical sequence in the story itself, then you could use the higher result as an initiative marker. But then again, I can imagine some initiative results being hard to interpret, such as if I rolled a 4 on a d4 (a success), and you rolled a 5 on a d10 (a miss), then your "miss" would come before my "success"? What would that look like? Maybe only hits would count? Misses would automatically go last?
In any case, I think it's worth exploring for certain implementations of a Powered-by-the-Apocalypse game, but you'd have to think carefully about the intention and what effect you want the die mechanic to have on play. These are just some of the ways I see, but you might see something else :)