[JENKINS-69815] Introduce a cache for private mode check

[rsandell]

Description

Another attempt like #698
See
JENKINS-69815 for further information.

Submitter checklist

• Link to JIRA ticket in description, if appropriate.
• Change is code complete and matches issue description
• Automated tests have been added to exercise the changes
• Reviewer's manual test instructions provided in PR description. See Reviewer's first task below.

Reviewer checklist

• Run the changes and verify that the change matches the issue description
• Reviewed the code
• Verified that the appropriate tests have been written or valid explanation given

Documentation changes

• Link to jenkins.io PR, or an explanation for why no doc changes are needed

Users/aliases to notify

[basil]

Merge with #684 at https://github.com/basil/github-branch-source-plugin/tree/JENKINS-69815

[jglick]

All this complexity is so we can decide whether AvatarMetadataAction should provide a logo for the org folder? #532 by @Vlatombe IIUC. Better to just delete this feature if it is causing problems, or set an icon once when you create an org folder, or offer some button in the config page to download and apply an icon, anything. This is just not important enough to be computing during retrieveActions.