Car Key Fob doesn't transmit RF, althougth everything else seems to be working
I am thinking it might be the coil. If the coil works fine, using the multimeter, measure its resistance on its two 'legs'/points. It should be something like 1 ohm, 3 ohms, 5 max. (Measure the resistance with the battery disconnected or you might damage your multimeter)
Check some videos with people repairing wireless car keys online, it might help:
https://www.youtube.com/watch?v=RRtFrShU-mk