Your regular soldering station will also contain a microcontroller, same as a shocking amount of other tools. They can be incredibly cheap (a couple cents if you have enough volume), so even if you could do it with analog circuits they can be a reasonable solution. And of course this soldering iron comes with buttons and a display, so of course you use a microcontroller.

But I guess what you are getting at is that normally you don't brag about the architecture of your microcontroller. As you correctly suspect, that's done just for coolness points. Chances are an 8 bit AVR or an 16 bit PIC could have done the same thing, but on a soldering iron you don't care about the power draw of the microcontroller, and a 32bit RISC-V fits Pine64's brand better. It's the equivalent of "aircraft-grade aluminum" or "military-strength encryption".

Also the choice makes it easier to run Doom on it.

Regular soldering stations don't contain microcontrollers, they use a thermocouple to decide if the circuit should flow through the iron or not and some other smaller stuff to smooth out the temp gain / ensure the overshoot doesn't get too wild.

You don't. I can recommend my daily driver, a naturally aspirated charcoal furnace and several large blocks of copper.

A PID for keeping temperature.

To let it turn off when you're not using it. It uses an inertial sensor.

That doesn't explain it, though. You don't need a microcontroller for that. I suspect the real answer is "to minimize parts count".

Is there a fixed-function circuit that can read a digital value out of a sensor and compare it with a threshold?

I'm not dounting it's possibly, but I'd be surprised if it's available without involving a Turing-complete computer.

Yes, there are tons of these sorts of circuits. It's something that is often needed, and these sorts of circuits have been used before ultra-cheap microcontrollers were a thing. A turing-complete computer is serious overkill for this sort of application.

I'm not sure if there is a single-chip solution (although I'd be a bit surprised if there isn't), but the fundamental circuit isn't that ambitious. Using a very low-end microcontroller is likely less expensive, though.