a) There are plenty of people around here that would argue that every single hit ever contains some element of physical damage. I don't share this view, but openly stating the opposite would cause some to abandon any such game as "not D&D". (I would presume, given comments that I've read here and elsewhere.)
b) I don't think you (quite) got what I was saying. I'm suggesting that maybe Sleep should do 3d6 Drowsiness damage, with the caveat that it can only drop targets unconscious. Charm Person should do 1d6/level charm damage, and if that drops the target, it switches to your side, otherwise it just saps its will to fight you. Weapon attacks and spells that do "normal" damage, would function similarly to each other, perhaps causing more substantial wounds if they "drop" you. If HP are really only "plot-token" staying power, then use them as such.
To be clear, I'm only musing on the idea. For all the reasons @the _Choice mentioned and a few more.
Strangely enough that's similar to how Aetherianica (the game I'm making right now) works.
You have three different hit point pools: Trauma, Stress, and Endurance.
If your Trauma hits 0 you are 'dying', if your Stress hits 0 you are 'having a mental breakdown' and if Endurance hits 0 'you are exhausted'. Each type of attack deals one of those kinds of damage. For instance a charm spell might deal Stress damage and when you reach 0 you are then under the control of the person that dropped you to 0. Endurance can be sapped by slamming on the shield and armor without dealing physical damage, eventually you are so battered and tired that you drag yourself across the battlefield barely able to swing your weapon.
You'd be surprised how flexible it really is.