Remove MongoDB room/result persistence and temporary dual writes

[coder13]

Part of #199.

Goal

Complete the PostgreSQL migration after the authoritative cutover has remained healthy for the agreed rollback window.

Scope

• Remove MongoDB reads and writes for rooms, race sessions, attempts, solves/results, and their participant state.
• Remove embedded attempt/result data and room racing fields from the Mongoose model.
• Remove the PostgreSQL dual-write implementation and flags that only supported MongoDB authority.
• Remove shadow-read comparison code and migration-only metrics after preserving any generally useful health metrics.
• Retain historical migration and reconciliation tooling only when it remains useful and safe.
• Update backup, restore, development, deployment, and architecture documentation.

Safety requirements

• Confirm no supported rollback image requires MongoDB room/result data.
• Advance any rollback floor before destructive cleanup.
• Capture and verify a final pre-cleanup backup.
• Make destructive MongoDB cleanup a separate, explicit, idempotent operation.
• Do not delete historical PostgreSQL attempts or solves.
• Verify production no longer queries the removed collections or fields before cleanup.

Acceptance criteria

• Application code contains no MongoDB room/session/attempt/solve reads or writes.
• PostgreSQL failures correctly fail affected durable operations rather than silently falling back.
• Migration-only dual-write and shadow-read flags are removed.
• Cleanup can dry-run, apply, verify, and rerun idempotently.
• A repository search and runtime instrumentation find no removed-path usage.
• Backups and restore instructions treat PostgreSQL as authoritative for these domains.
• Development and test environments no longer require MongoDB for room racing behavior.
• Normal and competition room end-to-end tests pass after MongoDB room data is unavailable.

Dependencies

• Successful PostgreSQL cutover and completion of its rollback observation window.