Expose recorder long-term statistics via REST API (parity with WebSocket)

[h3x89]

Describe your core improvement

Add a REST HTTP endpoint for querying long-term statistics (LTS) stored
by the Recorder integration — equivalent to the existing WebSocket command
recorder/statistics_during_period.

Proposed endpoint:

POST /api/statistics_during_period

accepting the same parameters as the WebSocket command: start_time, end_time,
statistic_ids, period, types.

Current limitations

Long-term statistics are accessible only via WebSocket API
(recorder/statistics_during_period). The REST API covers only short-term
state history via /api/history/period/ — limited by purge_keep_days
(typically 10 days). There is no REST path to LTS data that persists
indefinitely in the statistics table.

A related issue (#131355) was closed as stale with no resolution — a user
explicitly asked "Any way to retrieve the long term statistics through the api?"
and the answer was "maybe via WebSocket".

Technical benefits

• Parity between REST and WebSocket APIs for historical data access
• External tools that use the REST API (scripts, monitoring tools, AI agents,
  MCP proxies) gain access to LTS without needing a WebSocket client
• No new data schema needed — response mirrors the existing WebSocket format
• Implementation is straightforward: the WebSocket handler already exists,
  a REST wrapper would call the same underlying recorder function

Additional context

Concrete use case: LiteLLM MCP proxy exposes HA as a tool to an AI agent
via REST OpenAPI spec. The agent reads current states and recent history
just fine — but any trend analysis older than 10 days is impossible because
LTS is WebSocket-only. Workarounds (direct SQLite query, proxy sidecar) are
fragile and not portable.

REST API docs for reference: https://developers.home-assistant.io/docs/api/rest
WebSocket command docs: https://developers.home-assistant.io/docs/api/websocket