Mr. Heater Gas Valve Problem

Your "fix" sounds exactly like what I've experienced. As I said earlier these systems operate "on the edge." A partially blocked orifice changes the flame characteristics and "I guess" the flame temperature. I've run into this many times. No, the nut mounting the thermopile does nothing, it only needs to be snug. All electrical is done through the 2 wire conductors.

You might examine the stat and see if it has open, exposed contacts. If so, put that in the back of your head, and maybe "next time" replace the stat as well. the stat on mine is "newer" electronic, and uses a small internal relay which I can hear when it cycles. It's been in the system since about 2010

The stat MUST be rated for millivolt systems. In the older days, a typical manual stat for 24V systems had what is called a "heat anticipator" which is essentially an adjustable rheostat in series with the contacts. Millivolt stats have no such