Sorry for the late reply. I’m probably forgetting some of the technical names for things, but it went okay.
Yes, I arrived with visa-exempt status. After a number of phone calls, emails and interviews, I found a good position.
The school applied for a local work permit (county, I believe). Once approved, I was given paperwork to use to apply for an ARC at the BOCA office in Taipei. I filled out an application online, as well, submitted it online and printed out a copy as a backup.
Although I was anticipating an interview, I’m not sure the BOCA officer said or asked anything. The paperwork from the school, the local government and my application seemed to be the critical items. Everything went smoothly. It took perhaps 10 minutes as she made copies, stamped papers and so on.
I chose the option of having the ARC mailed to my school, to avoid another trip to Taipei. That meant a little running around in the same building, but it seems like they’ve developed a pretty efficient operation there. My ARC arrived 10-14 days later.
My only mistake was misunderstanding how long my visa-exempt status was good for. I thought it was 2 months and booked my required return flight accordingly. As an American, I actually had 90 days. To make everything match up on the application (and to give myself more time if something went awry), I shifted my flight and paid the $200 fee. Then I was able to put my accurate return flight info on my ARC application. They didn’t want to verify any of that, in person, but I wanted everything correct.
I’ve been here for 9 months and the job is going well.
My thanks again to @tempogain, @yyy and others who chimed in with advice. Let me know if I can add any more details to the above, @globalgourmand.