fix: the broadcast diagnostic reported NO-BINDING from a parse that never ran

The run I asked for came back with three alarming lines:

  lifeCycle=? privacy=? recording=? bound=NONE boundMatches=NO-BINDING
  autoStart=unset autoStop=unset liveChatId=none | streamStatus=active

Every single broadcast field was empty, and the stream half was fine. That
combination is the tell: not a failed request, a wrong nesting level.

liveBroadcasts.list returns a LIST wrapper, so the broadcast is at items[0]. I
descended into items for the stream and read the response root for the
broadcast, so every lookup missed, and BoundMatches — handed a null it could
not distinguish from a real null — reported a confident NO-BINDING. The
diagnostic invented the bug it was written to find. Three lines of fabricated
alarm on the one run that was supposed to produce evidence.

Two fixes, and the second is the one that matters:

  FirstItem() descends into items[0] for both halves, and returns default when
  there is genuinely no item.

  The line now prints boundMatches=unread when the broadcast could not be
  read at all, so an absent parse degrades to "I don't know" instead of a
  verdict. A diagnostic that guesses is worse than no diagnostic: this one
  would have sent me off to rebuild a binding that may well be correct.

Test guards the regression: FirstItem_DescendsIntoTheListWrapper_AndReports-
AbsenceHonestly, including the empty-items and missing-items cases. Suite
390/390, 0 warnings.

Credit to the run for catching it. What it did establish, from the half that
worked: streamStatus goes ready -> active about 30s into the push. So the
ingest registers as active server-side, which combined with the 99MB muxed
yesterday means the stream half is verified healthy end to end. The broadcast
half needs one more run to say anything.

ytLive.Tests/BroadcastBindingTests.cs

@@ -1,3 +1,4 @@
+using System.Text.Json;
 using Xunit;
 using ytLive.Services;
 
@@ -30,4 +31,28 @@ public class BroadcastBindingTests
     {
         Assert.Equal(verdict, YouTubeStreamService.BoundMatches(bound, expected));
     }
+
+    /// <summary>The regression the first diagnostic shipped with. `liveBroadcasts.list`
+    /// returns a LIST wrapper, so the broadcast sits at `items[0]`; reading the response
+    /// root reads the wrapper, every field comes back empty, and the binding check then
+    /// reports a confident `NO-BINDING` from a parse that never happened. The first real
+    /// test run produced exactly that — three lines of fabricated alarm. `FirstItem` must
+    /// descend, and must return `default` (not throw, not an empty object) when there is
+    /// genuinely no item, so the caller can report "unread" instead of guessing.</summary>
+    [Theory]
+    [InlineData("{\"kind\":\"youtube#liveBroadcastListResponse\",\"items\":[{\"id\":\"bc1\",\"status\":{\"lifeCycleStatus\":\"ready\"}}]}", "bc1")]
+    [InlineData("{\"items\":[]}", null)]
+    [InlineData("{\"kind\":\"youtube#liveBroadcastListResponse\"}", null)]
+    public void FirstItem_DescendsIntoTheListWrapper_AndReportsAbsenceHonestly(string listJson, string? expectedId)
+    {
+        using var doc = JsonDocument.Parse(listJson);
+        var item = YouTubeStreamService.FirstItem(doc.RootElement);
+
+        if (expectedId == null)
+        {
+            Assert.Equal(JsonValueKind.Undefined, item.ValueKind);
+            return;
+        }
+        Assert.Equal(expectedId, item.GetProperty("id").GetString());
+    }
 }