Hello Andy,
I'm not sure, but do you have now a Pro Subscription ?
If yes, i think that for some reasons, you can't give me an admin access to check all this... (if it changes, so open a pro private ticket with access, so that it would be easier to check where the issue could be :
pro.joomlic.com/en/support/individual-issues ;-) )
But, i think you have maybe a plugin, which is overriding the profile/user system of Joomla.
So, what could be done is to check if not-core joomla plugins are disabled, you still have the issue.
The only thing i can say is that is works for me with same settings as yours, on a Joomla install and iCagenda with no other third-party extensions.
And in addition, as i visit everyday user's sites, and can confirm that many user have private access for some forms, but have no issue with auto-logout (in the same time, this could be run only by a third-party extension such as plugin, as iCagenda itself does not run auto-logout of users).
Hope you will find where the issue is ?...
Best Regards,
Cyril