feat(api): #288 wrap raw VMs, nullable-honest DTOs, search pageNum

24 #nullable enable flips across ErsatzTV.Core/Api; Collection/Schedule/
SmartCollection/Resolution VMs wrapped in ResponseModels (Version now
header-only, SPA-verified); pageNum threaded into GET /api/search.

Refs #288 #197

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

ErsatzTV.Core/Api/Scheduling/ProgramScheduleResponseModel.cs

@@ -0,0 +1,16 @@
+#nullable enable
+using ErsatzTV.Core.Scheduling;
+
+namespace ErsatzTV.Core.Api.Scheduling;
+
+// API response DTO for a program schedule. Mirrors the Application ProgramScheduleViewModel's
+// config fields but omits the header-only optimistic-concurrency token (issue #253, #288): the
+// schedule-items GET/PUT carry Version as an ETag header, so the SPA never reads it from the body.
+public record ProgramScheduleResponseModel(
+    int Id,
+    string Name,
+    bool KeepMultiPartEpisodesTogether,
+    bool TreatCollectionsAsShows,
+    bool ShuffleScheduleItems,
+    bool RandomStartPoint,
+    FixedStartTimeBehavior FixedStartTimeBehavior);