I don’t know enough about the fediverse or Lemmy code to say how easy or hard this would be to implement but if we’re logging in with the username noogs@lemmy.noogs.me for example, it’s reasonable for the app to assume the server is located at lemmy.noogs.me and it can derive the likely URL of https://lemmy.noogs.me from that. The only case this wouldn’t work is if your instance is running on a port other than 443 because then we need some way to tell the app what the port should be.
Email (or at least Microsoft Exchange email) uses a protocol called autodiscover for this which uses DNS to tell an email client where to get connection information from, it then polls that URL for the information and configures the email client automatically. Using a similar DNS based approach may be useful as well.
I use my laptop for anything that requires a real keyboard or bigger screen. Then I have my server, and my phone. So I mostly just use the laptop and the phone. I do have a dual screen phone though so that helps a bunch for multitasking.