Reporting back for anyone who consults this thread in the future.

23 hour stopover in Doha (landing from SFO was scheduled for 1740) worked very well. Recommended!

Checkin automatically through-routed our bags; luckily we were prepared and had a change of clothes in our hand baggage. It was nice not to have to wait for bags.
Inbound flight was late, followed by absurdly long walk on and on and on with no travellators to help; then there was a VERY long queue at DOH immigration -- if travelling business class show your boarding pass and there is a whole separate immigration area for you, which was a HUGE timesaver even if the agent was taking 4 minutes for each person in their queue. Without that we might have spent an hour or more in the queue, a bit like JFK on a bad day (but calmer, without JFK's shouting officials).

Quick taxi into town - drop hand baggage at the Al Mirqab hotel (part of the Souq Waqif hotels). You get an hour or more to wander the interesting souq in the slightly cooler evening air, seeing the camels, horses, falcons etc - before going back to the hotel for a massage or hammam at 2100 or out to dinner in the souk.
Next morning, after breakfast, off to the Museum of Islamic Art. This is an architectural masterpiece and a very interesting collection.
We ran out of energy otherwise we might have had a short visit to the National Museum of Qatar. That will wait for a future visit.
Taxi back to the airport is only 15-20 mins.

Definitely recommended.