By "Create Pull Request button in the branches drop-down" I mean the drop-down beside the "Start the active scheme" button at the top of the XCode window. My project has both a local and a remote repository. Actually I have tried with more projects and I do not have the "Source Control > Create Pull Request..." menu item. I also have full access to the repository since it is my repo.

Well, this (Source Control > Create Pull Request...) is what's not available for me. (Also, the "Create Pull Request" button in the branches drop-down is missing.) And I can confirm that my XCode version is "13.1 (13A1030d)".