You don't even need a central server for that. Just set up a p2p network and have all peers broadcast their messages (¶) to everyone they're connected to. Those peers in turn forward everything they receive to everyone they are connected to and so on. (And, of course, they try to decrypt everything in the meantime to see if any message is meant for them.)
(¶) I think you could do without sending random data unless network consists of only two active peers and your adversary taps every single internet link. (As in, even the cable underneath the street in front of your house.) If he doesn't, broadcasting your message to your (physical) neighbors would likely be enough to obfuscate the message's origin.
Anyway, the idea of broadcasting / flooding to protect metadata is not new. See, for instance, Bitmessage and Ethereum Whisper. If you ask me, though, (and you mentioned that, too) the traffic and performance issues originating from broadcasting and brute-force decryption make this approach rather impractical to pretty much impossible – at least if you want this to work for the masses. (Even if it's just for text messages, not cat pics or anything.)