DataConnect field _embed requires ADC to be set up with gcloud #8819

[REQUIRED] Environment info

firebase-tools: v14.9.0

Platform: macOS 15.5

Firebase Data Connect extension: v1.5.0

[REQUIRED] Test case

https://github.com/aalej/fdc-embed-adc-mcve

[REQUIRED] Steps to reproduce

Using the test case

1. Run gcloud auth application-default revoke
2. Run gcloud auth revoke
3. Run firebase login
4. Run firebase use --add PROJECT_ID --alias default
5. Open the Firebase Data Connnect extension
6. Click the "Start emulators" button
7. Open ./dataconnect/RestaurantEmbedd_insert.gql, then click "Run (local)"

[REQUIRED] Expected behavior

The VS Code extension makes an ADC using the credentials from firebase login and the mutation succeeds

[REQUIRED] Actual behavior

The mutation errors with

failed to build http client for calling Vertex. Error failed to set up an OAuth token source. Error google: could not find default credentials. See https://cloud.google.com/docs/authentication/external/set-up-adc for more information