The “Continue in the browser” is greyed out for new accounts which has no buckets yet. It is not a web browser issue.
As I said, the “Continue in the browser” button becomes available (not grey out) after creating a bucket in the the “Object Browser” dashboard section. It was just unclear that buckets creating is hidden under that tab.
Since my main issue was that I was not able to find where to create a bucket to be able to create an access grant, the main question is solved.
As for buckets creating tab, you can ignore my technical issues because I have too aggressively restricted Firefox profile with “about:config” modifications. As for creating buckets at Pale Moon, I see
TypeError: self.newPermission is not a function error message showed on the EU1 page. Anyway, I do not need any browser because I have API key and can use the CLI.