I have already opened a port request in case anyone else is interested. There is a Drupal 7 module https://www.drupal.org/project/bawstats that uses the AWStats open source library, https://www.awstats.org/ and the module is an alternative to Google Analytics without extra intrusion as it analyzes the server log.
It looks like it should also work on shared hosting based on the resolution identified in this issue, https://www.drupal.org/project/bawstats/issues/1181880 that was fixed and closed. If anyone else is interested they can take a look at the port request https://github.com/backdrop-ops/contrib/issues/731 and add their comments to it.
Comments
Lots of us in Backdrop use Matomo which is open source and can be privacy friendly. There is a module for that and it is possible to install a Matomo instance on shared hosting.
Yes, Matomo is an option and one we use on other sites. The awstats is an option that allows seeing everything that is happening on the server without having to login to the server panel and the app is open source.