client-api: Rewrite websocket loop

crates/client-api/Cargo.toml

@@ -48,9 +48,11 @@ uuid.workspace = true
 jsonwebtoken.workspace = true
 scopeguard.workspace = true
 serde_with.workspace = true
+async-stream.workspace = true
 
 [target.'cfg(not(target_env = "msvc"))'.dependencies]
 jemalloc_pprof.workspace = true
 
 [dev-dependencies]
 jsonwebtoken.workspace = true
+pretty_assertions = { workspace = true, features = ["unstable"] }