I would say yes, this should include the local instance as well, although to be honest this FEP is written with Mastodon in mind, where only S2S is provided, so the actor is only represented for other servers.
I agree the name of the attribute is not ideal. The intent is really to provide consent for the purpose of generalized search.
There was not. There was a discussion of a per-object flag, but that was ruled out as unpractical if a user were to change their mind.
The reason, is, again, rooted in Mastodon’s history: from the moment search features were introduced back in 2017, they were purposefully limited to not have generalized search. This means that (many) Mastodon users have that expectation, and treating it otherwise would suddenly make their existing posts searchable by default without their explicit consent. We understand that different projects have different expectations, and that is why this is a SHOULD and not a MUST.