[2021] rust+ app not working

Make sure it's version 0.0.13 and try again. There were server issues since the patch came out but it's fixed now.

You will need notifications to be enabled and working to pair servers with your phone.

Which login screen are you talking about? The Steam login screen or the loading screen that says "Logging in"?

The loading screen shouldn't take more than a few seconds - if it is then try switching to wi-fi or data.