I think I would hate SP spending being capped 3 per turn, because it would increase the mental load, without giving enough in return. But if I (have to) try it out, I might change my mind.
Part of the issue (with the current system) might be that mana-based characters can spend SP to pick up those mushrooms, with Dex-based characters that defeats the point. Part of the issue is that mana-based characters use up SP too, so the shrooms refill SP and mana, which feels better. Part of the problem is that I hardly ever use up all my mana, and I'm often ok going into a fight at 80% mana, but I do often use up all my SP, and I really don't want to start a fight without maxed out SP. Part of the issue (or maybe the culmination of all this) is that mushrooms don't feel effective enough at refilling SP, when they do feel effective at refilling mana. I can't rule out it playing a role that the mushrooms are purple, not blue.
As for why waiting at all feels bad, having to press 'rest' too much slows the game down. And I do like that, honestly, it makes the game feel more alive.
The 'problem' you seem to diagnose, that SP regen would be 'too strong' could be fixed simply by reducing the amount of SP regen. Maybe 0.2 / Dex / turn is too strong, then just pick a smaller number. 0.1 regen every other level would let you reach half a sprint point per turn at the cost of not getting 80 hp.
Getting max hp increases is very valuable to me, with most characters it's worth putting at least 2 points into Strength. I don't see that changing, unless the numbers associated with Dex are too high. (But I can't see the point in investing in Int with Dex- or Str focussed characters, unless it is to buy specific talents. That wouldn't change either.)
As for the equipment, look you can't guarantee getting +maxSP equipment in any particular game. I agree there should be equipment in the game which grants extra maxSP, whether that is by explicitly granting +maxSP or implicitly by granting Dex doesn't matter (to me). None of my or Justin-Wang's proposals would have removed +maxSP form all these items. I think you're making a mountain of a mole-hill here.
If I'm reading you correctly you really like the current system because getting 2 extra SP (from Dex or equipment) is (for you) extremely valuable, and at some point they stop being so valuable even for bow users, and that encourages characters to hybridise, and you want to keep that?