I'm not extremely knowledgeable about this, so take this all with a grain of salt, but i'll write some nonsense that comes to mind (also, I pretty much respond to any notification i see - especially from npubs i've known for a while - and I don't get enough to not see them):

- wider reach without murdering the edge device. the remote caching server could potentially connect to hundreds of remote relays and your edge device only connects to this single remote caching server.

- this one is contentious, but: filtering. depending on whether you think this is good or bad, a centralized server can use a lot of user statistics and behavior to filter spam or otherwise "undesirable" content (personally not a fan)

- ordering/promotion: largely similar to above. the cache server can determine what order you see posts in, how often you see notes from a given user, etc. etc. (personally not a fan)

- heavy computation: like nostr:npub12262qa4uhw7u8gdwlgmntqtv7aye8vdcmvszkqwgs0zchel6mz7s6cgrkj alluded to recently, the "caching server" (really stretching this term now) can perform high-resource or long-running computations that your client can simply read "for free"

- probably other stuff but those come to mind are are probably most relevant to the wider discussion at the moment.

For the ones I'm "not a fan of", I think those general patterns are very good things, but I don't think we have the proper WoT and personal server infrastructure yet to achieve it in a way that doesn't move us backwards in terms of decentralization and content moderation/censorship. These are the sorts of features that everyone wants for good reason, but if "we give it to them" in a half-assed way, I believe we diminish the chances of ever achieving it in the full-assed way. (mostly due to user complacency. the internet repeats this cycle over and over)