IMCE -> Security notice: ...will not save backup files to the server because the destination directory is publicly acces

This issue is just to document a problem that might occur in a particluar situation and what causes it, so that the next person who runs into it can find the answer.

Using IMCE module (7.x-1.5)
In settings fior IMCE (admin/config/media/imce)

unchecking this setting:
Disable serving of private files
IMCE serves all files under private files directory without applying any access restrictions. This allows anonymous access to any file(/system/files/filename) unless there is a module restricting access to the files. Here you can disable this feature.

Throws the error: Security notice: Backup and Migrate will not save backup files to the server because the destination directory is publicly accessible

So just don't do that, it is a bad idea anyway. Marking this as "won't fix"