Why won’t the propane boiler’s pilot stay lit after replacing the thermocouple

I actually have one of these and there is a known issue with the Honeywell gas valves used. The solution (if it applies to your model) is to replace the valve.

It sounds to me like a restriction at the regulator, not the control valve. You mentioned that the previous owner had done some work on it, make sure that he/she did not replace the regulator with one that does not deliver the design flow/pressure. In this case, adequate flow/pressure depends on proper regulator back pressure setting (they are rated in "inches of water" back pressure). If the regulator setting is too low, it will not deliver an adequate flow/pressure of gas.

If this is the case, the initial proper operation followed by diminishing performance is explained by the volume of gas in the 8' tall hose assembly, with a slow reduction in adequate volume being delivered by the undersized regulator. I had your exact problem when I attempted to swap out a leaky patio heater regulator with a BBQ regulator that had a different pressure setting.

Like this:

• Regulator is initially open because of low (zero) downstream pressure.
• 8' tall hose fills with gas until rated back pressure downstream of regulator is reached, regulator closes.
• You open control valve and light heater, regulator opens as downstream pressure drops.
• Damaged, defective, clogged, or undersized regulator cannot supply enough gas to maintain proper flame height as pressure drops, but does stay open and deliver gas according to it's pressure setting, which is not enough for adequate flame propagation.

It's worth checking, anyway.