engines don't like being shock cooled, so you need power on to keep the engine warm. An extended period at idle when continuing at flying speed will damage an engine
"Shock" and "Extended period" are rather opposed terms. Shock cooling is bad, by degree of rate of temperature change. Allowing the engine to cool gradually will not create a problem for the engine.
If you carry power through the flare, you are cheating yourself out of the landing gear warning on most RG types, that could become hard to explain if you forgot.
The techniques for flying on the back side of the drag curve do not work well with the techniques for landing in significant crosswinds. If you have that short a runway, and that big a wind, another runway is probably a better decision.
Step Turn tell us he happily flew an aircraft from the maintenance facility that did dodgy brake maintenance to another location rather than have that facility fix it
Well... not "happily", I was rather put out to be honest. But, the aircraft was not
at a maintenance facility, the mechanic had left aircraft the day before at the private runway. I was taking it
to the maintenance facility at the destination.
But, I agree, students, don't fly unserviceable planes....