kiding
Save the user password and auto-login or auto-fill.
#7 can stay signed-in but never login automatically. A keychain access is required to store and retrieve the username and password.
There's this convenient API for keychains called SecKeychainFindInternetPassword. Sadly after a few tests it seems the app can't access iCloud keychain.
Perhaps a quick fix would be to check if the user clicks the Facebook or Twitter button to log in, and if so, "click" this programmatically when the apps starts. This seems like a temporary, but easily to implement, fix.
Clicking seems easy, but I think detecting if the user is logged in with Facebook or Twitter would take somewhat more time.
After #7 gets in place, there will be much less login screen to be seen. Let's decide if auto-fill (which only fills the username and password field) is just enough or auto-login (auto-fill + triggering login) is still needed after #7 is closed.