The only correct kind of faucet is the kind with a thermostatic handle for temperature, and a separate handle for flow.

They are almost exclusively available for showers and that's stupid. It's the correct solution for the problem, with extremely few drawbacks.

The only improvement I could see is making the temperature handle short circuit to the drain until the water going through is hot enough. But I dont know if that's really possible without electronics, and I really dont want my faucet handles to be plugged in or take AA batteries.

There are! They’re called thermostatic faucets.

We have mixer taps in our bathroom sinks and showers. Just lift the handle up and the temp is the same as it was last time you used it.

It's reasonably common for showers to have a mixing valve and a flow rate valve on separate handles. That accomplishes what you want. You just have to remember which is which and only use the flow rate valve to turn on and off.

More importantly, hot water circulation systems should be more common. It's the waiting for the cold water in the line to flush out that really makes setting temperature a hassle.

My kitchen faucet does this. It's a 2-axis lever. Y axis is the temperature adjustment, X axis is flow. As long as you leave it set to the same Y position when you turn it off and on, it'll be at the "last used combination".

My super basic faucet handles are exactly that…you twist it left-right to set temp and tilt it up-down for pressure from off to full. We just leave it rotated wherever we like it for temp and tilt up to turn on each time to the desired pressure. Our water pressure is always variable, so the amount tilted up varies, but the “rotational temp” almost never needs changing. There’s no fancy thermostatic valve in these like some shower have. There’s even those fancy kitchen sink faucets that remember everything and you just tap them with your hand and they automatically turn on/off to your settings of temp and pressure. I think they are called “touchless faucets”. Pretty sure even Ikea sells one.

I feel like sink handles like mine are super common, too. I’ve had similar ones in the states and in Europe…

Indeed :). That's what these are.

would love indexing and numbers on my handle

or just go full Japan and incorporate a thermostat

You mean a (single handle) mixer tap?

Keep in mind that mixing the levels of hot and cold water isn't the only factor in the final temperature. It's also the actual temperature of the water in the pipes. Depending on where your pipes run, the cold water in the pipes may be warmer or cooler than the underground source of the water. The hot water may also have cooled more or less since leaving your hot water heater. Initial temperature may therefore be too hot or cold compared to where it ends after a period of use.

Have you tried using your own memory?

Something like this works. Just turn the water off with the spray nozzle, not the handle. https://share.google/1HpxlLwZ6RMXGdMoS

I used to have a kitchen sink faucet with the mixing handle on the side, that i installed after the 2 tap mixers.

So the 2 taps controlled hot and cold, which fed into the faucets cold line, and the hot line was capped off, making the handle only control flow, and temp was the 2 knobs. (I did this out of laziness because I didnt want to unsolder and resolder in new fittings)

I bet it would not be hard to 3d print a little ring with a twisty arrow that you could use to "save" your last handle placement, kinda a neat idea.

everytime

Not a word, my dude.