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