No start

If it had anti-theft, it would run for a few seconds until the fuel pressure bled down. The place to start is by checking for stored diagnostic fault codes. You can do that by cycling the ignition switch three times from "off" to "run" within five seconds without cranking the engine, then the codes will be displayed on the odometer. If there are no codes, you'll need a scanner that can display live sensor data. Look at the cam and crank sensors to see if "present" or "no" is displayed. If both are displayed as "no", it's possible no code will be set in memory because the Engine Computer looks for a signal from one when it sees the signal from the other one. If both signals are missing, the computer doesn't perform those self-tests.

If both signals are missing, you'll have to check the common 5, 8, or 10 volt feed wire and the common ground wire.

You're a few steps ahead of most people with your dandy observations of the ASD relay.

Burned contacts in the ignition switch typically affect the circuit feeding the power windows, heater fan, and radio. The contacts for the engine circuit handle very little current. Passing the large current is the job of the ASD relay.

Code 12 means the battery was disconnected within the last 50 starts. If it was not, look for an intermittent electrical connection that would make the Engine Computer think it was. Code 55 just means "end of message". Any other codes that might have existed would have been erased if / when the battery was disconnected or run dead. If no other codes come back, you'll have to use a scanner to view live data to see what's going on. The Chrysler DRB3 scanner has a "no-start" menu that shows all of the pertinent information on one screen. Normally when the ASD relay doesn't turn back on during cranking, the problem is related to the cam or crank sensors, the common feed and ground wires, or a timing chain or belt problem.