"They said they did have a problem because of the Tesla's automatic systems (shutting down the car when it was exited, IIRC), but they worked around it by having somebody sit in the driver's seat while the work was being done."
Exactly what you said here.
One reason some shops have problems. The work around is it requires 2 people. I believe there is a procedure for enabling the steering while out of the vehicle, possibly a service mode option. Some shops are cheap and don't want to pay 2 people do do one alignment and refuse.
Reminds me of when I wanted to put the windows down and listen to music outside the vehicle without it shutting off. I had to reach inside (after the door was shut), push down on the seat to get the screen to come back on, then turn up the music.
One of these days hopefully all shops will figure it out.