Serve /api/config endpoint with managed connections

## Parent PRD

#1660

## What to build

Add a `GET /api/config` endpoint that returns the client-relevant subset of the config file (everything except the `server` block). This replaces the current `/defaultConnection` endpoint and `defaultConnection.json` file generation.

The endpoint returns connections (keyed by stable ID with name, url, queryEngine, authentication, fetchTimeout, nodeExpansionLimit), feedbackUrl, and any other client-relevant fields. The `server` block is filtered out since it contains server-only settings.

## Acceptance criteria

- [ ] `GET /api/config` returns client-relevant config as JSON
- [ ] `server` block is excluded from the response
- [ ] Connections include their stable key/ID in the response
- [ ] Returns empty connections object when no config file is present
- [ ] Old `/defaultConnection` endpoint removed
- [ ] Integration tests for the endpoint with various config states

## Blocked by

- Blocked by #1661

## User stories addressed

- User story 1
- User story 7

## Related Issues

- Part of #1660

Start by tracing the existing /defaultConnection endpoint and defaultConnection.json generation, then review how the config file and managed connections are represented. The work is done when GET /api/config returns client-relevant JSON with stable connection IDs, excludes server, handles missing config with empty connections, removes the old endpoint, and has integration coverage for the stated config states.

Written by the indexing model from the issue text.