Fix OpenAPI servers field missing when root_path_in_servers=True (#12246)

When root_path_in_servers=True (default), the OpenAPI schema should always
include a servers field. Previously, when no root_path was configured (or
root_path was '/'), the servers key was entirely absent from the OpenAPI
JSON output. This fix ensures:

- servers=[{"url": "/"}] when root_path_in_servers=True and no custom
  servers or root_path are configured
- servers=[{"url": "<root_path>"}] when root_path is set
- servers are omitted only when root_path_in_servers=False
- Custom servers are preserved and root_path is prepended when applicable

fastapi/applications.py

@@ -1104,7 +1104,14 @@ class FastAPI(Starlette):
             async def openapi(req: Request) -> JSONResponse:
                 root_path = req.scope.get("root_path", "").rstrip("/")
                 schema = self.openapi()
-                if root_path and self.root_path_in_servers:
+                if self.root_path_in_servers and "servers" not in schema:
+                    if root_path:
+                        schema = dict(schema)
+                        schema["servers"] = [{"url": root_path}]
+                    else:
+                        schema = dict(schema)
+                        schema["servers"] = [{"url": "/"}]
+                elif root_path and self.root_path_in_servers:
                     server_urls = {s.get("url") for s in schema.get("servers", [])}
                     if root_path not in server_urls:
                         schema = dict(schema)