Thanks for the video demo, I think you can contact PS support and check the issue with login using your Gmail account on the addons marketplace.
If the back office shows the login page again or a blank screen after entering the username and password
Try changing the PHP version from your web hosting Cpanel. It worked for me in the past.
Attached is a screenshot for your reference.
Also, you can try different browsers or private windows in the same browser. (Edge, Firefox, Chrome, Safari, Opera etc)