Which type of client would it be though? The best possible version for what use case?
It makes sense for a protocol to have a dedicated web page for developers and people who want information on the protocol. The protocol likely won't have just the social media use case, so making the protocol website a client (I assume you mean social media client) doesn't make sense in my opinion. It's possible to have clients that use Nostr in ways that don't necessarily look like social media (like a music only website, or something like Medium for publishing). Client UI/UX can use Nostr without the bits that make the client 'social.'
It could make sense to have a section on the protocol website that highlights the most popular use cases though. The protocol team could even create/endorse an official web client for the social media use case and link to it on the protocol website. That makes more sense to me.
As an example, I have visited Bitcoin.org for information on the Bitcoin protocol/project. Having it open a web wallet instead would not make sense, but having links to an official web wallet would (like the Monero website).