diplomail (Stage C): paid-tier broadcast + multi-game + cleanup

Closes out the producer-side of the diplomail surface. Paid-tier
players can fan out one personal message to the rest of the active
roster (gated on entitlement_snapshots.is_paid). Site admins gain a
multi-game broadcast (POST /admin/mail/broadcast with `selected` /
`all_running` scopes) and the bulk-purge endpoint that wipes
diplomail rows tied to games finished more than N years ago. An
admin listing (GET /admin/mail/messages) rounds out the
observability surface.

EntitlementReader and GameLookup are new narrow deps wired from
`*user.Service` and `*lobby.Service` in cmd/backend/main; the lobby
service grows a one-off `ListFinishedGamesBefore` helper for the
cleanup path (the cache evicts terminal-state games so the cache
walk is not enough). Stage D will swap LangUndetermined for an
actual body-language detector and add the translation cache.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

backend/internal/server/contract_test.go

@@ -167,6 +167,18 @@ var requestBodyStubs = map[string]map[string]any{
 		"subject": "Contract test admin subject",
 		"body": "Contract test admin body",
 	},
+	"userMailSendBroadcast": {
+		"subject": "Contract test paid broadcast",
+		"body": "Contract test paid broadcast body",
+	},
+	"adminDiplomailBroadcast": {
+		"scope": "all_running",
+		"subject": "Contract test multi-game broadcast",
+		"body": "Contract test multi-game broadcast body",
+	},
+	"adminDiplomailCleanup": {
+		"older_than_years": 1,
+	},
 }
 
 // TestOpenAPIContract is the top-level OpenAPI contract test. It