Update 1 - not only AirPods, my Bose bluetooth headphones also having same issue. Can't get any Bluetooth headphones to playback, no matter if iOS tells me if they are connected or not.
Update 2 - Erase all content and settings has cleared the issue, now restoring from back-up to see if that helps.
Update 3
- Restored from backup and exactly the same issue, can't select AirPods as playback destination....hmmm.
Update 4
- Did a “reset all content and settings” then setup from scratch (no restore from backup) and all is now working fine, this has also fixed a bug where pull down to search on homescreen wouldn’t find all apps (for example shortcuts wouldn’t show).
Update 5 - Reading a few threads, it appears that some folks are seeing this problem occur when sound recognition is turned on, the fix is to simply turn off this feature (wish I'd found this last night to avoid the re-setup of my iPhone, but oh well that's beta life).