Remove WebSocket wire format (#10179)

I'd like WireApi to go away (when chat is removed) and WebSockets is
still responses API just over a different transport.

codex-rs/core/tests/suite/client.rs

@@ -1151,6 +1151,7 @@ async fn azure_responses_request_includes_store_and_reasoning_ids() {
         stream_max_retries: Some(0),
         stream_idle_timeout_ms: Some(5_000),
         requires_openai_auth: false,
+        supports_websockets: false,
     };
 
     let codex_home = TempDir::new().unwrap();
@@ -1671,6 +1672,7 @@ async fn azure_overrides_assign_properties_used_for_responses_url() {
         stream_max_retries: None,
         stream_idle_timeout_ms: None,
         requires_openai_auth: false,
+        supports_websockets: false,
     };
 
     // Init session
@@ -1751,6 +1753,7 @@ async fn env_var_overrides_loaded_auth() {
         stream_max_retries: None,
         stream_idle_timeout_ms: None,
         requires_openai_auth: false,
+        supports_websockets: false,
     };
 
     // Init session