Unable to view logs unless running as a local remote MCP server

I wanted to see additional information during deployment and any logs that were produced. By following the [steps here for "local" setup](https://github.com/GoogleCloudPlatform/cloud-run-mcp?tab=readme-ov-file#use-as-local-mcp-server), I wasn't able to see any additional logs.

Maybe we should write to a file if running locally? Is there another way to view this output?

Here's what I had to do to view logs:

1. edit the gemini_extension.json to use `"url": "http://localhost:3000/sse"` (note: i had to tweak something in the GCP_STDIO check - see https://github.com/GoogleCloudPlatform/cloud-run-mcp/issues/113)
3. from local terminal 1, `gcloud auth login`
4. from local terminal 1, `gcloud auth application-default login`
5. from local terminal 1, `GCP_STDIO=false node /cloud-run-mcp/mcp-server.js
6. from local terminal 2, launch `gemini` and ask it to deploy a service

then I could see logs in terminal 1.

Start with the local setup instructions and mcp-server.js, then inspect how GCP_STDIO and the gemini_extension.json URL affect output. Reproduce the deployment using the two-terminal steps in the issue and define a consistent way for local users to view deployment logs, with the result verified during a local run.

Written by the indexing model from the issue text.