Does this chat protocol exist?
I don't know of any chat protocol that would use this approach.

I get it from your question that once the symmetric key is established, Alice and Bob communicate directly using it and that it is not an issue that the server, if malicious, can defeat any security achieved by this.

However, it appears to me that your protocol suffers from a major problem, namely the fact that the server itself is not authenticated in any way. That means that an adversary with control of the network could impersonate the server, which by design knows the secret symmetric key Alice and Bob would be using.

Alice and Bob also never authenticate each other and rely on the lower layer and the supposingly secure shared key to communicate, so they may be impersonated very easily in such a scenario.