Must be a short somewhere. I'd normally suggest to check the battery connections, but if lights were working, then that couldn't be it.
Are you sure the battery is still good? If the computer doesn't think it has enough amps to start the motor, it won't let it turn over at all (lights will still work). Maybe sitting on the tow truck and slightly warmer temps got the resting voltage just above the threshold to start. The charging system would then keep voltage high enough as long as it doesn't sit too long between starts.
I'd check resting voltage (12.4-12.6 depending on temp), or ideally take it to an auto parts store so they can load test it.
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