Earlier quoted context omitted.
github just removed git: urls from their web ui so it no longer tells you how to make a clone you can push from using standard git it's been replaced with instructions for using their own gh cli tool (which, however, is also free software) for example on any github project page, https://github.com/username/projectname , click on the "code ↓" button in the top right; the "local" tab on the left has a "clone" tab with…
You have to be logged into github for the SSH clone tab to appear-- if they don't know your pubkey you can't connect using the SSH protocol. Git can push over HTTPS, too.
whether i'm logged in in my browser has nothing to do with whether my ssh client has access to an authorized private key