feat(ui): per-kind active-game limit lock on the New Game screen

Carry the caller's per-tier active-game caps and each game's kind on the
wire (Profile.game_limits + GameView.kind, additive FBS + gateway transcode
+ client codec, committed regen). The New Game screen counts the player's
active games per kind from the lobby and locks a capped start: an outline
button with a lock that opens a funnel modal instead of a game -- a sign-in
prompt for a guest, a "finish a current game first" notice for a signed-in
account (native Telegram popup, in-app modal elsewhere). The lock lifts via
the existing profile refetch after a guest->durable upgrade.

Remove the lobby's old at_game_limit New-Game tab disable + notice: the flag
(now the random-kind cap) conflicted with the per-kind lock -- it hid the
screen where the lock lives and wrongly blocked an unfulfilled kind. The New
Game tab is always enabled; the per-kind start lock is the only gate. The
at_game_limit wire field stays (unused by the client) for a later cleanup.

Tests: client lock logic + codec kind/game_limits roundtrip + gateway
transcode encode + native popup builders (unit); a mock e2e for the lock
badge and the modal.

docs/ARCHITECTURE.md

@@ -666,11 +666,14 @@ in either direction (the enqueue excludes the caller's `BlockedWith` set);
   refuses a **guest** outright with **403** (guests cannot use friends — the same guest gate
   covers friend requests and friend-code redemption). **Accepting** an invitation
   (`POST /invitations/:id/accept`) is never gated, so friend games are capped only at
-  initiation. The lobby learns the random-kind state from a boolean **`at_game_limit`** on
-  the `games.list` response — already re-fetched on entry and every game event; the client
-  additionally counts the listed games per kind against the per-tier caps it reads in the
-  profile, locking a kind's New-Game entry (a login funnel for a guest, an already-at-limit
-  note for a durable account) when that kind is full.
+  initiation. The client counts its lobby games per kind against the per-tier caps it reads on
+  the profile (`Profile.game_limits`, carrying `GameView.kind`), and locks a capped kind's
+  New-Game start — a 🔒 outline button that opens a prompt instead of a game: a sign-in funnel
+  for a guest, a "finish a current game first" notice for a durable account (native inside
+  Telegram, an in-app modal elsewhere), lifting when the profile is re-fetched after a
+  guest→durable upgrade. The `games.list` `at_game_limit` boolean (now the random-kind cap) is
+  still carried but no longer drives the UI — the per-kind start lock superseded the old
+  New-Game-tab disable.
 - **Friends**: two add paths over one `friendships` table. A **one-time
   code** the to-be-added player issues (a `friend_codes` row: 6-digit numeric,
   SHA-256-hashed, **12 h** TTL, one live code per issuer, single-use, redeem