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/UI_DESIGN.md

@@ -347,12 +347,15 @@ its entrance (`components/Toast.svelte`, re-keyed on a per-message sequence). An
 lives ~2 s, drifting up by about the tab-bar height as it fades (a plain fade, no travel, under
 reduce-motion); an **error** toast dwells longer (~4 s). Either dismisses instantly on tap.
 
-**Simultaneous-game cap.** When the player is at the active-quick-game cap (`games.list`
-`at_game_limit`), the lobby's **New Game** tab is **disabled** (greyed via the shared
-`.tab:disabled` opacity) and a plain, low-emphasis line — muted, centred, no accent — sits
-under the lists with the localized "limit reached" notice. Both follow the flag carried in the
-cached lobby snapshot, so they render in their last-known state on entry (the button stays
-enabled on the first, uncached load) and flip in place when an event refreshes the lobby.
+**Active-game caps.** A player's simultaneous unfinished games are capped per kind (vs AI / random /
+friends) against the caller's per-tier limits (`Profile.game_limits`), counted client-side from the
+lobby games. The lobby's **New Game** tab is **always enabled** — the lock lives on the per-kind
+start, not the tab. On the **New Game** screen a start whose kind is at its cap renders as an
+**outline** button with a **🔒** (not the filled accent CTA), so it reads as gated rather than ready;
+tapped, it opens a short prompt instead of a game — inside Telegram a **native popup**, elsewhere the
+in-app **Modal**. A **guest** sees a sign-in funnel (Cancel / Sign in → the account screen); a
+signed-in account at its cap sees a plain "finish a current game first" notice (OK). The lock lifts in
+place when the profile is re-fetched after a guest→durable upgrade.
 
 ## Social, account & history surfaces