feat(client): include gateway presence in diagnostics

PLAN.md calls for user-facing diagnostics that can say whether the remote
client is connected to the LAN gateway. Startup already reported the initial
welcome bit, but periodic diagnostics only carried relay reachability and route
pinning.

Add gateway_connected to RelayDiagnostics and seed the client status from the
welcome. The client control-event logger now updates that status when gateway
join and leave events arrive, so later diagnostics reflect relay lifecycle
changes while the tunnel is running.

Test Plan:
- cargo fmt --check
- cargo test -p lanparty-obs -p lanparty-client-win
- cargo test --workspace
- cargo clippy --workspace --all-targets -- -D warnings
- git diff --check

Refs: PLAN.md

crates/lanparty-obs/src/lib.rs

@@ -196,14 +196,16 @@ impl TunnelStats {
 pub struct RelayDiagnostics {
     reachable: bool,
     route_pinned: bool,
+    gateway_connected: bool,
 }
 
 impl RelayDiagnostics {
     #[must_use]
-    pub const fn new(reachable: bool, route_pinned: bool) -> Self {
+    pub const fn new(reachable: bool, route_pinned: bool, gateway_connected: bool) -> Self {
         Self {
             reachable,
             route_pinned,
+            gateway_connected,
         }
     }
 
@@ -216,6 +218,11 @@ impl RelayDiagnostics {
     pub const fn route_pinned(&self) -> bool {
         self.route_pinned
     }
+
+    #[must_use]
+    pub const fn gateway_connected(&self) -> bool {
+        self.gateway_connected
+    }
 }
 
 #[derive(Debug, Clone, PartialEq, Eq, serde::Deserialize, serde::Serialize)]
@@ -428,7 +435,7 @@ mod tests {
         let mac = MacAddr::new([0x02, 1, 2, 3, 4, 5]);
         let stats = TunnelStats::new(1, 2, 3, 4, 5, 6);
         let diagnostics = ClientDiagnostics::new(
-            RelayDiagnostics::new(true, true),
+            RelayDiagnostics::new(true, true, true),
             QuicDiagnostics::new(true, Some(1400)),
             TapDiagnostics::new(
                 true,
@@ -441,6 +448,7 @@ mod tests {
 
         assert!(diagnostics.relay().reachable());
         assert!(diagnostics.relay().route_pinned());
+        assert!(diagnostics.relay().gateway_connected());
         assert!(diagnostics.quic().datagram_supported());
         assert_eq!(diagnostics.quic().max_datagram_size(), Some(1400));
         assert!(diagnostics.tap().adapter_found());