feat(admin): show the newest database dump on the backups page

GET /api/admin/backups now carries `database`: the newest good dump (when,
how big, kept on the server, off-host or both), the newest failed run,
whether this server can write its backup directory, and whether off-host
copies are configured. The backups card leads with one row built from it,
with a notice when the last run failed and when dumps can only go
off-host or cannot be taken at all. The route stays an unpublished admin
console route, so the OpenAPI document does not change.

src/types/backups.ts

@@ -9,6 +9,7 @@
  * both sides a route-handler-independent shared type home; the route
  * keeps owning the HTTP contract, the component keeps owning the UI.
  */
+import type { DatabaseBackupHealth } from "@/lib/database-backup/health";
 import type { BackupPassActivity } from "@/lib/jobs/backup-pass-attempts";
 import type { BackupScheduleStatus } from "@/lib/jobs/backup-schedule-status";
 import type { OffhostBackupFreshness } from "@/lib/jobs/offhost-backup-freshness";
@@ -101,6 +102,12 @@ export interface BackupsList {
    * the operator's bucket, and those two fail independently.
    */
   offhost: OffhostBackupOverview;
+  /**
+   * The nightly whole-database dump, the copy that rebuilds a lost server:
+   * the newest good dump (kept here, off-host, or both), the newest failed
+   * run, and whether this server can keep dumps at all.
+   */
+  database: DatabaseBackupHealth;
   /**
    * Soft retention hint — the worker is configured for weekly backups
    * (see `DATA_BACKUP_CRON` in `src/lib/jobs/reminder-worker.ts`), and