This was somewhat covered previously https://news.ycombinator.com/item?id=34221851 but is worth having a look at.
In short though, any attempts to do this tends to result in huge block lists, the hashtag #fediblock circulating around, and in some cases outright personal attacks on the people behind it.
Then a movement to defederate anyone who allows indexing.
Then hopefully a move to allow-list federation which I personally have suggested for a while now as the only sensible move if they really want some privacy.
(Checking for “web search tool”, #Google and #Fediblock from mastodon.social, it seems like people have not noticed yet.)
I would like to point out that HN has actually discussed the “no search social media” idea before, where it was called pseudo-privacy[1].
Whitelisting makes me worried for the future of decentralised social media, as higher barriers for creating new instances encourage consolidation instead of diversity and decentralisation. It’s true that it would make moderation easier, though. Indeed, one of the main barriers to spam right now is perhaps the cost of domains.
Speaking of which, this also makes me worried how the Fediverse will deal with its inevitable spam problem, without simply saying “the volunteer mods will take care of it”. Or perhaps that is a legit strategy, I’m not sure.
I have a feeling that Hashcash could be a solution, and where instances do not participate in it you could slow their connections in. Just an idea though.