(feature) Add staff-only service status API

Add /api/status/ for quick inspection of database backend, object counts, latest
backup run, and latest scheduler activity. Link it from the API index and reuse
schedule serialization between host summaries and status output.

Cover the endpoint with a focused API test and document the new status URL.

src/pobsync_backend/api.py

@@ -3,8 +3,10 @@ from __future__ import annotations
 from typing import Any
 
 from django.contrib.admin.views.decorators import staff_member_required
+from django.db import connection
 from django.db.models import Count
 from django.http import JsonResponse
+from django.utils import timezone
 
 from .models import BackupRun, HostConfig, ScheduleConfig, SnapshotRecord
 
@@ -15,6 +17,7 @@ def api_index(request) -> JsonResponse:
         {
             "ok": True,
             "endpoints": {
+                "status": request.build_absolute_uri("/api/status/"),
                 "hosts": request.build_absolute_uri("/api/hosts/"),
                 "snapshots": request.build_absolute_uri("/api/snapshots/"),
                 "runs": request.build_absolute_uri("/api/runs/"),
@@ -23,6 +26,35 @@ def api_index(request) -> JsonResponse:
     )
 
 
+@staff_member_required
+def status(request) -> JsonResponse:
+    latest_run = BackupRun.objects.select_related("host", "snapshot").order_by("-created_at").first()
+    latest_schedule = ScheduleConfig.objects.select_related("host").order_by("-last_started_at", "-updated_at").first()
+
+    return JsonResponse(
+        {
+            "ok": True,
+            "generated_at": timezone.now().isoformat(),
+            "database": {
+                "vendor": connection.vendor,
+                "engine": connection.settings_dict["ENGINE"],
+            },
+            "counts": {
+                "hosts": HostConfig.objects.count(),
+                "enabled_hosts": HostConfig.objects.filter(enabled=True).count(),
+                "schedules": ScheduleConfig.objects.count(),
+                "enabled_schedules": ScheduleConfig.objects.filter(enabled=True).count(),
+                "snapshots": SnapshotRecord.objects.count(),
+                "runs": BackupRun.objects.count(),
+                "running_runs": BackupRun.objects.filter(status=BackupRun.Status.RUNNING).count(),
+                "failed_runs": BackupRun.objects.filter(status=BackupRun.Status.FAILED).count(),
+            },
+            "latest_run": None if latest_run is None else _run_payload(latest_run),
+            "latest_schedule": None if latest_schedule is None else _schedule_payload(latest_schedule),
+        }
+    )
+
+
 @staff_member_required
 def hosts(request) -> JsonResponse:
     host_qs = (
@@ -78,14 +110,7 @@ def _host_payload(host: HostConfig) -> dict[str, Any]:
         },
         "schedule": None
         if schedule is None
-        else {
-            "cron_expr": schedule.cron_expr,
-            "enabled": schedule.enabled,
-            "prune": schedule.prune,
-            "last_status": schedule.last_status,
-            "last_started_at": _iso(schedule.last_started_at),
-            "last_finished_at": _iso(schedule.last_finished_at),
-        },
+        else _schedule_payload(schedule),
     }
 
 
@@ -143,6 +168,19 @@ def _run_payload(run: BackupRun) -> dict[str, Any]:
     }
 
 
+def _schedule_payload(schedule: ScheduleConfig) -> dict[str, Any]:
+    return {
+        "host": schedule.host.host,
+        "cron_expr": schedule.cron_expr,
+        "enabled": schedule.enabled,
+        "prune": schedule.prune,
+        "last_due_key": schedule.last_due_key,
+        "last_status": schedule.last_status,
+        "last_started_at": _iso(schedule.last_started_at),
+        "last_finished_at": _iso(schedule.last_finished_at),
+    }
+
+
 def _limit_from_request(request, *, default: int = 100, maximum: int = 500) -> int:
     value = request.GET.get("limit", str(default))
     try: