If you use the private mode exclusively it oughta do the same thing for you.

But Firefox's private mode brings new problems - such as not being able to use saved logins/passwords (as of today Nov 14, 2020).

Example: I browse New York Times, Amazon, Bank of America, and Reddit. I have logins for all of these. If I use non-private mode, they all get to set cookies and track me indefinitely after I login. If I use private-mode, I don't get to login to any of them.

It feels like that the app designers haven't thought through a lot of this.

Agreed. Private browsing is a complicated alternative to a problem that's easily solved in the desktop browser by configuring it to automatically delete all cookies when the browser closes except for those specified. Why isn't this feature available in the mobile version?