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.

gateway/internal/backendclient/api.go

@@ -47,6 +47,16 @@ type ProfileResp struct {
 	// DictVersions is the current dictionary version per game variant, forwarded verbatim
 	// into the Profile payload so an offline-capable client preloads the matching dawg.
 	DictVersions []DictVersion `json:"dict_versions,omitempty"`
+	// GameLimits is the caller's tier active-game caps per kind (-1 = unlimited), forwarded verbatim
+	// into the Profile payload for the client's per-kind New-Game lock.
+	GameLimits *GameLimitsResp `json:"game_limits,omitempty"`
+}
+
+// GameLimitsResp is the caller's tier active-game caps per kind (-1 = unlimited) in the profile.
+type GameLimitsResp struct {
+	VsAI    int `json:"vs_ai"`
+	Random  int `json:"random"`
+	Friends int `json:"friends"`
 }
 
 // AdsResp is the post-move interstitial config in the profile: the client-mirrored cooldowns
@@ -170,6 +180,7 @@ type GameResp struct {
 	Seats                []SeatResp `json:"seats"`
 	UnreadChat           bool       `json:"unread_chat"`
 	UnreadMessages       bool       `json:"unread_messages"`
+	Kind                 int        `json:"kind"`
 }
 
 // MoveResultResp is the outcome of a committed move. Rack carries the actor's refilled rack as