No. Nostr has no relationship to Bitcoin, does not depend on Bitcoin and no one has to have ever heard the word "Bitcoin" uttered to them in order to use Nostr.
Different approaches can be used, but the most common one is that your client will automatically announce the relays you generally publish to, and your friend's client will automatically connect to these relays specifically to fetch your posts (and connect to other relays to fetch other people's posts).
It is not true that Nostr doesn't have moderation or censorship. Nostr just doesn't have a central committee of censors, but each relay is privately owned and can censor anyone they want. As the network grow it will become more common for relays to have some form of sign-up process so not everyone can publish a million offensive notes to every relay using throwaway identities and get away with it.
But even in a world in which relays don't do any of that clients can still protect users from harassment by fetching only what they want from relays and filtering out garbage locally -- in fact many clients already do this using various heuristics.
This is not what happens. Relays do not have to store everything and are never expected to store everything. Relays can store posts only from people they want, and only for the time they want (they can delete things). A relay database can be as small as few kilobytes just for your personal notes.