Figured I'd update this thread: we rented a car and were very happy we did. Allowed us to explore the whole island at our own pace.

Being out of season, parking was never an issue. We stayed in Oia and parked at the Post Office. We went to Fira one day and parked in a free lot. In other towns we just parked wherever....on the side of the road. No issues with traffic, occasionally had to yield to let someone pass on a narrow road.

Of course rental cars are a stick shift but I'm used to that.

Only mild surprise: "SUV" is a bit of a misnomer. It was still a small car, but had a hatchback and a little more trunk space. It was a tight squeeze for four people with luggage, but comfortable enough once the luggage was offloaded.