You have it backwards! The thermostat closes the flap and keeps the hot air out. Without it, the flap springs open and hot air gets in all the time, so it will work great in the winter.

If you want to keep hot air out of the intake, leave the thermostat alone and tie the flap closed. You can take the thermostat out, if you want, but it is pointless as long as the flap is tied shut.

I may be wrong, but I don't believe that hot air can burn out the AMM. Here's why: there is an electrical current through the platinum wire which heats it to a constant temperature regardless of the air temperature. If the air is hot, it takes less current to heat the wire, but the temperature of the wire stays the same. The wire is hotter than the air in any case, otherwise the AMM wouldn't work.