It sounds like you are suggesting that the healer be able to stabilize the dying character and resolve that conflict with a single roll. That's not a bad way to change the power so it acts more like the angel kit. But by the book, that's not what the move does, you only heal a single level of harm.
Yes. A single level of harm can keep a character from dying. If that resolves what is interesting about the character being wounded, skip the rest. If he is probably going to heal, let him heal and pick it back up at that point.
I'm not sure what your point was with regard to different wounds taking different amounts of time. Are you suggesting that multiple levels of harm could be healed but the single roll might represent a longer period of time?
No, simply that the time frame of using the move isn't set. It can take six seconds or six hours. Players can only spam this move if the MC allows them time to do so. If it is taking screen time away from other players, keep the amount of rolls equal between them (in the case that spamming the move means hogging spotlight, of course--might not be an issue).
This is a little bit of a digression, but I was surprised that you thought that it is the MC's responsibility to frame scenes and decide their scope and range. A lot of the time I have seen the MC ask questions while the player frames the scene. The rules are pretty explicit about MC responsibilities, but I don't think this was addressed.
Aside from "Your job as MC is to say everything else" (p 109), no, certain basics of rpgs (like what a GM is) are not addressed. Only what makes AW unique. Still, the MC is both a Game Master, similar to other GMs but not the same, and a Master of Ceremonies. It's your job to preside over it, which includes keeping it moving. Although I was specifically referring to pp. 121-122, "A Few More Things to Do."
Also, I found it odd that you don't think we should be beholden to the wounds on the character sheet. If we're going to ignore them, why are we tracking them in the first place?
It isn't ignoring them. They are a mechanical representation of the fiction, which means they change to represent the fiction, when it changes. If Dremmer gets you with his shotgun, you take 3-harm. If the next scene is 6 months later, after you've healed up, you erase that 3-harm. Because your wounds have healed. But if a gunfight ends with you the sole survivor, having taken 5-harm, unable to walk, and ten miles from town, then it's probably important to see if you live or die in those moments. But if you heal your wounds in the fiction, you heal your wounds on the character sheet. All I'm suggesting there is that when it is no longer interesting to deal with those wounds, you should just skip ahead to when they are healed, instead of rolling healing touch a bunch of times.
Even so, there might still be times when rolling it a few times in a row might be a good idea mechanically, and still work fictionally. But I don't think you need to try re-writing the move just yet.