A thought for Visibility is to use user profiles like I described, but just check if email is empty or not from the user profile. It should only be filled if the user is signed in, so there is no need to check ‘is signed in user’ when you can simply check if the user profile email is filled.
Another thought to work around filters is you could probably set up your tabs in detail view, then on the sheet you are using for each tab, add a template column that gets the email from the user profile and uses it to build your relations to the relevant data using email for the relation, so you wouldn’t have to worry about setting filters by signed in user and could keep the app public.
You may have also seen this already as a hack to get around the signed in user issue on public apps, but it’s not the best solution.
However, I think the first 2 solutions may be something that would work better. I haven’t them yet other than a brief test, but it might work and it’s something that I will definitely have to try in one of my apps.
Any of these solutions should keep your app public. The third hack option does work, but I don’t know if it would work forever and any time you would want to add something like a filter that requires signed in user, you would have to change the apps privacy setting each time just to make the change.