Yea, I'm lost. You say you don't have a 'working key' and you keep talking about "syncing" a key. Syncing the key has to do with the lock/unlock function, it has NOT A THING to do with starting the car.

When you put the key in the ignition and turn to "on" the skreem reads the RFID chip in the key. If the chip responds properly, all is good.
If the chip does not respond, or the SKREEM has gone bad, you get the "three cranks and nothing" symptom, this is a designed behavior to tell you/the garage that the SKREEM is not giving the ECM the "ok" to start.

I think the SKREEM is bad, unless someone has had the dash apart and they accidentally left the cable from the "Key RIng" (my term) unplugged from the SKREEM. Or you have replaced your key Fobs and didnt transplant the RFID chip to the new FOB.

Not sure what the "Frying eggs" and "battery drain" thing is all about.....