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/config.schema.json

@@ -443,6 +443,11 @@
           "minimum": 0.0,
           "type": "integer"
         },
+        "supports_websockets": {
+          "default": false,
+          "description": "Whether this provider supports the Responses API WebSocket transport.",
+          "type": "boolean"
+        },
         "wire_api": {
           "allOf": [
             {
@@ -1081,13 +1086,6 @@
           ],
           "type": "string"
         },
-        {
-          "description": "Experimental: Responses API over WebSocket transport.",
-          "enum": [
-            "responses_websocket"
-          ],
-          "type": "string"
-        },
         {
           "description": "Regular Chat Completions compatible with `/v1/chat/completions`.",
           "enum": [