fix(games): carry game kind on live events + fix the New Game lock funnel

Address review of the active-game limit lock:

- Live-event GameViews (opponent_moved, match_found, game_started, ...) did
  not carry game_kind, so a lobby patch from an event zeroed a game's kind
  and the client's per-kind count under-counted — a capped kind read as
  free (a disabled start instead of the lock, and a wrong per-kind result).
  Thread Kind through notify.GameSummary → the event GameView.

- New Game screen refreshes the lobby games on mount so the per-kind count
  reflects the current set, not a stale cached snapshot.

- The guest funnel's login button routes to the profile screen (the account
  controls), not settings.

- Copy: the guest prompt is "sign in to use all the game's features"; the
  durable notice is "finish your active games to start a new one".

Regression tests: the notify opponent-moved payload carries kind; the client
lock locks only the reached kind (vs_ai at cap leaves random open).

ui/src/screens/NewGame.svelte

@@ -18,7 +18,7 @@
   import { validDisplayName } from '../lib/profileValidation';
   import { verifyPin, type PinLock } from '../lib/pin';
   import { commitName, rosterReady, buildSeats, shuffleSeeded, type RosterRow } from '../lib/roster';
-  import type { AccountRef, Variant } from '../lib/model';
+  import type { AccountRef, GameView, Variant } from '../lib/model';
   import {
     availableVariants,
     VARIANT_FLAG,
@@ -60,7 +60,10 @@
   // funnel modal instead of enqueuing (the server also refuses it); offline never locks (local
   // games are uncapped). The lock lifts when the profile is re-fetched after a guest→durable upgrade.
   const autoKind = $derived(opponent === 'ai' ? GameKind.vsAi : GameKind.random);
-  const autoLocked = $derived(!offlineMode.active && isKindLocked(getLobby(false)?.games ?? [], app.profile?.gameLimits, autoKind));
+  // The player's current games for the per-kind lock: seeded from the lobby snapshot for an instant
+  // render, then refreshed on mount (below) so a game created since the last lobby visit is counted.
+  let lobbyGames = $state<GameView[]>(getLobby(false)?.games ?? []);
+  const autoLocked = $derived(!offlineMode.active && isKindLocked(lobbyGames, app.profile?.gameLimits, autoKind));
   let limitOpen = $state(false);
 
   // --- auto-match ---
@@ -129,6 +132,16 @@
   );
 
   onMount(async () => {
+    // Refresh the lobby games so the per-kind lock counts the current set — the cached snapshot can
+    // lag a game created since the last lobby visit. Online only; the lock never applies offline, and
+    // the server enforces every cap regardless.
+    if (!offlineMode.active && connection.online) {
+      try {
+        lobbyGames = (await gateway.gamesList()).games;
+      } catch {
+        // Keep the cached seed on a transient failure.
+      }
+    }
     // Offline mode offers only the local vs_ai flow (the friends section is hidden), and the network
     // is off, so skip the friend fetch — it would be refused by the transport and toast an error.
     if (guest || offlineMode.active) return;
@@ -349,7 +362,7 @@
           open={limitOpen}
           {guest}
           onclose={() => (limitOpen = false)}
-          onlogin={() => { limitOpen = false; navigate('/settings'); }}
+          onlogin={() => { limitOpen = false; navigate('/profile'); }}
         />
       {:else if offlineMode.active}
         <!-- Offline pass-and-play (hotseat): a device-local 2-4 human game. The host sets a master