feat: Expose game state as Home Assistant entities (round, leader, score, active)

[mholzi]

Feature Request

Expose key game state information as Home Assistant entities so users can build dashboards and trigger automations based on live game events.

Suggested entities:

• sensor.beatify_current_round — current round number
• sensor.beatify_leader — name of current leader
• sensor.beatify_top_score — current top score
• binary_sensor.beatify_game_active — whether a game is currently running
• sensor.beatify_player_count — number of active players

Use cases:

• Show a live game dashboard in Lovelace
• Trigger light scenes when a game starts/ends
• Automate announcements via TTS when someone takes the lead
• Log game sessions to history

UI / Admin Discovery

The admin should be made aware that these entities exist and how to use them. Suggested additions to the admin UI:

• Info banner or section in the admin panel (e.g. Settings or About tab) listing all available Beatify entities with their entity IDs
• A short note like: "Beatify exposes live game state as Home Assistant entities. Find them under Developer Tools → States, or search for beatify in your entity registry."
• Optionally: a direct link or button that opens the HA entity registry filtered by the Beatify integration

This ensures admins discover the feature without having to read the docs or check the integration manifest manually.

Originally suggested in #440 by @Marcovandervlag