Batch alternate version detection in DtoService to remove MediaSourceCount N+1

Browsing a page of videos with the MediaSourceCount field ran one alternate
version query per item, each opening a fresh DbContext. On a large library that
turned a single page into hundreds of sequential round trips and made the Items
endpoint take tens of seconds while holding a request thread the whole time.

Detect which videos own alternate versions once per page with a single query,
mirroring the existing people batch. Videos absent from that set have a single
media source, so the per item lookups are skipped for the common case. Behavior
is unchanged: a video with no alternates already resolved to a count of one.

Adds a regression test asserting the count resolves from the batch and the per
item lookups are never called.

Jellyfin.Server.Implementations/Item/LinkedChildrenService.cs

@@ -59,6 +59,27 @@ public class LinkedChildrenService : ILinkedChildrenService
             .ToArray();
     }
 
+    /// <inheritdoc/>
+    public IReadOnlySet<Guid> GetItemsWithAlternateVersions(IReadOnlyList<Guid> itemIds)
+    {
+        if (itemIds.Count == 0)
+        {
+            return new HashSet<Guid>();
+        }
+
+        using var dbContext = _dbProvider.CreateDbContext();
+
+        var parentIds = dbContext.LinkedChildren
+            .Where(lc => (lc.ChildType == DbLinkedChildType.LocalAlternateVersion
+                    || lc.ChildType == DbLinkedChildType.LinkedAlternateVersion)
+                && itemIds.Contains(lc.ParentId))
+            .Select(lc => lc.ParentId)
+            .Distinct()
+            .ToArray();
+
+        return parentIds.ToHashSet();
+    }
+
     /// <inheritdoc/>
     public IReadOnlyDictionary<string, MusicArtist[]> FindArtists(IReadOnlyList<string> artistNames)
     {