Assuming you don't want to make the current repo public, you have to rename the current repo and create a new public repo with your username. Obviously that will break redirects, so update your existing references to that repo.

Pretty sure it has to be a public repo with a README.md in it

Put the README in it?

The secret trick is to create a new public repository and name it your 'username'. That should do it.

Also nice to see this new hack using GitHub actions to keep the current updated. Here is my profile README (kept it simple), love to see others as well: https://github.com/suyashjoshi