Did my car cover cause the NATS system to completely "fry"

The cover wouldn't have caused that issue. More likely is that the car isn't used much and the switch developed internal corrosion as a result. Do you put a battery tender or trickle charger on it to keep the battery up? I ask because those NATS systems can trigger as a result of low voltage or being jump started.
I'm not a fan of covers myself, depending on your location and where the car is stored they can cause paint damage as they are put on and off, If stored outside they can develop condensation between the cover and the car that can damage the paint as well. Plus the biggest issue is that they cover the car and create a nice dark area that rodents like. Then you end up with chewed wiring or them nesting inside various areas.