[Bugfix] Honor stage selection in online profiler endpoints (#6609)

Signed-off-by: qujing226 <qujing226@gmail.com>

tests/entrypoints/openai_api/test_api_server_guards.py

@@ -1,5 +1,5 @@
 # SPDX-License-Identifier: Apache-2.0
-# SPDX-FileCopyrightText: Copyright contributors to the vLLM project
+# SPDX-FileCopyrightText: Copyright contributors to the vLLM-Omni project
 """API-server surface guards for the OpenAI Omni entrypoint.
 
 Purpose
@@ -20,8 +20,8 @@ Common wrong changes these tests are meant to catch
 * Route accidentally removed, renamed, method-changed, or double-registered
   after a move (census / assembled-app checks).
 * OpenAPI drops an Omni HTTP path while the router still has it (or the reverse).
-* Assembly forgets to strip upstream ``/v1/chat/completions`` (batch) /
-  ``/v1/models``, or also deletes unrelated upstream routes.
+* Assembly forgets to strip overridden upstream chat / models / profiler
+  routes, or also deletes unrelated upstream routes.
 * Profiler / Omni router not mounted; storage not started; Omni engine
   exception handlers not registered.
 * Timestamp middleware stops stamping HTTP, or starts mutating WebSocket scopes.
@@ -351,9 +351,10 @@ def test_router_openapi_paths_cover_http_manifest() -> None:
 async def test_api_server_assembly_replaces_upstream_routes_and_mounts_omni_router(monkeypatch) -> None:
     """Lock worker assembly: override, mount, storage, handlers, full census.
 
-    Fails if assembly stops replacing upstream chat/batch/models, deletes
-    unrelated upstream routes, forgets Omni/profiler mounts, skips storage
-    start, or drops Omni ``EngineDeadError`` / ``EngineGenerateError`` handlers.
+    Fails if assembly stops replacing upstream chat/batch/models/profiler,
+    deletes unrelated upstream routes, forgets Omni/profiler mounts, skips
+    storage start, or drops Omni ``EngineDeadError`` / ``EngineGenerateError``
+    handlers.
     """
     captured: dict[str, object] = {}
 
@@ -372,6 +373,14 @@ async def test_api_server_assembly_replaces_upstream_routes_and_mounts_omni_rout
         async def upstream_models():
             return {"owner": "upstream"}
 
+        @app.post("/start_profile")
+        async def upstream_start_profile():
+            return {"owner": "upstream"}
+
+        @app.post("/stop_profile")
+        async def upstream_stop_profile():
+            return {"owner": "upstream"}
+
         @app.get("/v1/upstream-only")
         async def upstream_only():
             return {"owner": "upstream"}
@@ -437,6 +446,8 @@ async def test_api_server_assembly_replaces_upstream_routes_and_mounts_omni_rout
         is api_server.create_batch_chat_completion
     )
     assert _single_http_route(routes, "GET", "/v1/models").endpoint is api_server.show_available_models
+    assert _single_http_route(routes, "POST", "/start_profile").endpoint is api_server.start_profile
+    assert _single_http_route(routes, "POST", "/stop_profile").endpoint is api_server.stop_profile
 
     # Assembled app includes Omni router HTTP/WS surface + profiler.
     for item in _EXPECTED_ROUTER_ROUTES: