That's definitely what the only existing caching server does. But I think you can have a meaningful cache that covers only part of the network, depending on the users it serves.

Are you referring to Primal? And if so, does it/can it cover the whole network? Can't someone have their own private relay or npub that only they can see?

Yeah, of course, but they at least attempt to get full coverage. You make a good point, private indexing/search is useful and can't be accomplished by big providers