Can confirm that you cannot use the app with logging in with a Reddit account.

Just yesterday, I got tired of the annoying "use the app or login" messages when tapping to view more comments on a post, so I caved and installed the app - given the language of the nag, I thought I wouldn't have to login, and the Reddit UX of constant full-page reloads just to view more comments is such a joke I figured the app had to be better.

But no, you have to login with a Reddit account to use the app :/