The "myTrailhead Branding Management" permission referenced in the error message in your screenshot is the same permission as "Manage Site Settings" (it was renamed recently).
If your user has this permission enabled, I would expect you to be able to log in.
Try logging in with a different browser or an incognito window. If you're logged into other Salesforce services on other tabs, sometimes the Trailhead login doesn't quite keep up with the changing/updating contexts. It may also be a caching issue.