Surprise! The AirPods paired with my iPhone. From start to finish, here's what happened.
1) unpair AirPods from iPhone
2) follow the procedure to pair AirPods (AirPods pair as "not your airpods")
3) pairing dialog hangs
4)cancel pairing
5) try to pair AirPods
5a) AirPods status light flashing white
5b) pairing dialog doesn't appear when case lid opened
5c) device doesn't appear in bluetooth menu
6) reset iPhone settings
7) attempt to pair AirPods
7a) AirPods status light flashing white
7b) pairing dialog doesn't appear when case lid opened
7c) device doesn't appear in bluetooth menu
7d) attempt paring on an iPad linked to the same iCloud account (no pairing dialog, no device in bluetooth menu)
8) give up
9) come back 3 hours later
10) open case lid, see status light flashing green
11) close AirPod case lid
12) unlock iPhone
13) open AirPod case lid, press and hold pairing button until status light flashes white
14) pairing dialog automatically appears
15) AirPods pair, become usable
The fact that two separate devices refused to acknowledge the existence of my AirPods tells me that the issue was either account related (as both devices were linked to the same account) or AirPod firmware related.