What works

• Configuring the Jenkins organization folder ("Configure"), changing a setting and clicking "Save". 

The org scanner is immediately triggered and it finds every new branch, new PRs (including PRs from forks), triggers jobs for new pushes.
NOTE: This magic only works after changing an important setting (eg I selected a different Credential.  Just changing Description did not help) ** 

• Manually triggering the scanner at the individual repository sublevel. (not org level)

What doesn't work

• Scheduled scan (eg every 1 minute): it finds nothing new. 

• Manually triggered scan ("Scan Organization Now"): it finds nothing new.

Versions

Same behavior with:

• Jenkins 2.289
• Jenkins 2.269
• GitHub branch source 2.10.2
• GitHub branch source 2.9.2 

Note that we are currently switching from Bitbucket to GitHub Enterprise, and this worked normally with Bitbucket (also self-hosted).

Regardless of whether GitHub changes are detected, the scanner output always looks the same.  Only one branch is ever in the output (I have many, plus PRs):

Proposing testrepo Examining
My-Org/testrepo
Checking branches... Getting remote branches... Checking branch
main
‘Jenkinsfile’ found Met criteria 
1 branches were processed (query completed) 
1 branches were processed 
Finished examining My-Org/testrepo