🧹 Extract database initialization check into a utility method (#289)

Extracted the repeated `!document.databaseOperations` check and `document.databaseOperations` accessor into a new private `ensureDatabaseInitialized` utility method in `HostBridge` to reduce duplication and improve maintainability.

Co-authored-by: google-labs-jules[bot] <161369871+google-labs-jules[bot]@users.noreply.github.com>

Author: google-labs-jules[bot]