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

まず既存の /defaultConnection エンドポイントと defaultConnection.json の生成をたどり、次に config ファイルと管理対象の接続がどのように表現されているかを確認します。GET /api/config がクライアントに必要な JSON を安定した接続 ID 付きで返し、server を除外し、config がない場合は空の接続として扱い、古いエンドポイントを削除し、指定された config 状態に対する統合テストのカバレッジを備えた時点で作業は完了です。

索引モデルが issue の本文から書いたものです。