I think it’s crucial for people to realize that Fediverse has basically no privacy protections, so regardless of what SHOULD be, fact is you don’t even have the choice.
When you post to Fediverse there is zero guarantee that your content won’t be subject to search.
It’s just part of the design of the system, for better or worse.
Fediverse was built around giving instances all the power, so if an instance wants to search, it can.