I wanted to update my GOG Downloader, but the update page is empty (apart from the standard gog.com frame). Anything broken there?

This is where "Check for updates" takes me to:
https://www.gog.com/en/downloader/0.9.0.1376/

I think that means you already have the latest version. The 0.9.0.1376 version has been around at least as long as you have been a member here, so I don't see how you could have an out of date version.