No, if the problem was that a mouse got caught in the fan, it was the condenser fan that wasn't working; not the evaporator fan. Most fridges don't have a condenser fan and rely on convection for cooling the compressed refrigerant. That's what that black tubing at the back of a typical fridge is all about. It's generally larger fridges like side-by-sides that have the condenser coil at the bottom of the fridge and a fan to blow air through that condenser coil. Smaller fridges typically have their condenser coil on the back of the fridge.

I expect that a condenser fan that's gone on the blink for whatever reason would affect the fridge's ability to keep food cold and frozen, but I've never heard of a fridge overheating and being ruined because of the condenser fan not working. I expect that appliance manufacturers test their fridges to make sure the condenser coil is large enough to allow for sufficient cooling even if the condenser fan isn't working.