Also, this leaks unnecessary metadata. If you know Alice's address, and you know that she talked to Bob at 8:05 yesterday, you can figure out Bob's address by following the money.
If we want to use a gossip protocol, users will have to run federated servers and that requires a different incentive system. In this system, users pay a small fee and are assured that their message reaches the intended party.
New addresses can be used for new chats, since these addresses can be generated randomly, there is no way to connect them to an identity. Money can be routed to the address through a Bitcoin mixer to increase anonymity. (https://news.bitcoin.com/coingeek-to-fund-the-bch-anonymizer...)
That's a cool solution to the problem, but you didn't need attach bitcoin addresses to it at all. For example, in Bitmessage, the source address is inside the encrypted portion. You can tell how many messages someone received, but not where they came from. If you separate your routing from your anti-spam incentives, your protocol will be more secure.