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.

backend/internal/server/handlers_admin_limits.go

@@ -9,6 +9,16 @@ import (
 	"scrabble/backend/internal/gamelimits"
 )
 
+// gameLimitsDTOFor resolves the caller's tier active-game caps into the profile DTO, so the client
+// can lock a capped New-Game start per kind. It returns nil when the limits config is not wired.
+func (s *Server) gameLimitsDTOFor(isGuest bool) *gameLimitsDTO {
+	if s.gamelimits == nil {
+		return nil
+	}
+	l := s.gamelimits.LimitsFor(isGuest)
+	return &gameLimitsDTO{VsAI: l.VsAI, Random: l.Random, Friends: l.Friends}
+}
+
 // consoleLimits renders the per-tier, per-kind active-game limit form (backend.config), read from
 // the in-memory cache.
 func (s *Server) consoleLimits(c *gin.Context) {