he “site:” operator won’t work when searching for topics discussed e.g. on Lemmy since there is no common, general domain name.

Agreed, that’s the problem.

Maybe that would be a great next step for a search-engine provider to be better: Add a new, flexible operator, like scope, and index the data of the fediverse or other distributed services for search.

Also agreed, that would be a solution. I’m doubtful wether they deem a particular network important enough to justify the extra effort on their part.

Is it possible to pipe search queries through a website? If yes, then let’s make a fedi-search! In which you can find any publicly available fediverse content.

Would it then be possible to google for site:fedi-search?