The PT6 114A /3 blade Hartzell does produce less thrust in feather than at fine pitch ground idle at taxi speeds. I fly float planes, fine ground idle in nil wind will pull the aircraft through the water at around 6 knots. Feather ground idle will pull the aircraft through the water in the same conditions at around 2 knots. The surge in thrust as you feather is very noticeable on water - you have to plan for it.
I approach boats / moorings in feather in light winds and in beta in moderate winds and stronger. In light conditions a beta approach can blow the boat away from you, in moderate winds or above a feathered approach may not allow you to reach it.
As for cooling to stabilize temps for shutdown - it is the cooling air supplied by the compressor that cools and stabilizes the internal engine component temperatures - a PT6 gas generator can't tell if the prop is in feather or beta - increase the engine power though and watch the ITT rise. Do keep an eye on the oil temp. BTW feathering has no significant effect on fuel flow in a PT6 - they are a free turbine.
W