All the key fob battery does is run the little transmitter for the lock/unlock/panic buttons. THAT IS IT.

The RFID chip in the key is passively powered by the SKREEM when it is in the ignition and surrounded by that black ring (which is a loop antenna), so the battery in the fob has nothing to do with starting the car.

If the car won't lock/unlock with the fob, try your spare key. If the spare works, then change your fob battery.

If the car wont' start right, THAT would be an indication of the car battery being bad.