My electronics knowledge is NOT great ... so it is easily possible that I have an error ... so let's take a look at that part of the circuit. What we see is that with the Reset switch open, REST should be pulled high by R6. When the switch S1 is closed, C1 starts conducting until it is saturated. That brings REST low for a short period of time. Even if the switch maintains closed, once C1 charges, it will stop conducting and REST will go high again. When S1 is opened, R7 discharges the capacitor so that when next S1 is closed, the reset will happen again.
Now ... this is what I built for my environment and it worked ... however, that doesn't mean it is "correct" ... I may have gotten "lucky". Again, I'm not skilled in electronics and discrete components like capacitors and resistors even less (I'm ok with digital as I can just about think of it like a program in hardware).
I have been testing with ESP01 and ESP12s.
What happens if you take a wire and short directly to REST ... does it restart then?
Neil