and like if i had to design the magical perfect shower system, the two dials would not be 'hot water' and 'cold water' but rather 'temperature' and 'flow' that can be adjusted independently
This exists and is called a "thermostatic mixer" faucet. @nia--san has probably seen ones with digital screens, but the tech is completely mechanical and are usually sold without any electronics at all. They're also not expensive; here's one for USD$20.
Basically every Japanese shower works like this, and, I repeat, it has zero electronics:
As you can see, the right knob adjusts temperature, and the left knob adjusts pressure. There's also a button on the showerhead to turn it on and off, so once you set the temperature and pressure you like, you can just press the showerhead button before and after every shower.
I've seen these controls in a few luxury hotels (and airport lounges) outside of Japan, but it's pretty rare rare.
adjusted independently with small movements of the dial corresponding to small, predictable changes in the relevant physical variable
You can see from the pic that this one has a bunch of temperatures marked on the dial. You can literally just select the exact temperature you want.
(e.g. water famously has a pretty big heat capacity, so there's gonna be some inertia if you're trying to change its temperature quickly)
Not actually a problem. As you can see, the mixer bar has hot water and cold water going directly into it, so you're not changing the temperature of the water itself, so much as changing what percentage of the mixture is coming from hot water.
In my experience, feeling the change happens pretty quickly, but that's also true of every other shower I've used (which also work this way), so I don't know what's up with your showers.
I have no idea how the showers of rich people work but are they remotely like this
Nope, I've been to a lot of pretty rich people's houses and seen some ultra-luxury showers and they may be huge and made of marble and have rain showers, but the controls seem optimized for artistic value rather than convenience.
I'd much rather use a Japanese shower. It's not like thermostatic mixers are expensive (the first one I found on Amazon Japan was USD$20). They just haven't caught on in the rest of the world for whatever reason.