Please critique my high-current voltage regulator
A circuit does rely on component values so you really should have made those realistic. Your circuit cannot deliver 4 A at the output as R2 and R3 have far too high values. It would need more than 80 kV (80000 Volts) to make 4A flow through 10 k resistors. So I'm assuming R2 and R3 have a value of a few ohms which should be more realistic.

(1) will it work in hardware as expected (it works on simulation)?

Theoretically yes but it will be extremely inefficient as you're just "burning off" any excess current through M1 and the resistors. At 4 A this will get very hot especially when there is no load. Which is just silly and not needed as there are far better solutions available.

This circuit is like driving a car at full throttle all the time and controlling the speed with the brakes. How long will those brakes last? What will be the fuel consumption? Bad!

(2) how stable/reliable this design?

I do not dare to say as it is too unconventional. As it is an inefficient solution, components get hot, hot components means a lower reliability.

(3) maybe there are better solutions for a problem (like cheap ICs or modules)?

Yes, there should be Switching DCDC (Buck) converters which can fit your requirements.

(4) how do engineers usually solve such tasks (i.e. voltage stabilization/regulation under a high-current)?

By using a Buck converter. I advise not try to build your own as the component layout (on the PCB) and the component choice is somewhat critical. I advise to get a module. Here's one based on the XL4005 which should fit your needs. The XL4005 can work with only 0.6 V voltage difference. It is cheap and easy to use. Don't forget to adjust the output voltage to 5 V before connecting anything to the output. The small screw on the blue trimmer sets the voltage.

In general it is a waste of time to "design" something like this in a simulator as I (having 30 years of experience with circuit simulators) can make "almost anything" happen in a simulator. I mean make circuits that will never work in the real world.

A simulator is a tool which need to be used to confirm what a circuit designer designs. Without sufficient knowledge of circuits a simulator's result / behavior can send you into a completely wrong / unrealistic direction.