feat(websearch): settings UI overhaul and quota improvements

- Remove Priority field, auto load-balance by quota remaining
- Replace QuotaRefreshInterval (daily/weekly/monthly) with SubscribedAt
  (subscription date, monthly lazy refresh via Redis TTL)
- Add collapsible provider cards, API key show/copy, usage progress bar
- Add test endpoint (POST /web-search-emulation/test) bypassing quota
- Wire WebSearchManagerBuilder on startup (was never called before)
- Fix nextMonthlyReset day-of-month overflow (Jan 31 → Feb 28)
- Fix non-deterministic sort in selectByQuotaWeight
- Map ProxyID in builder for provider-level proxy tracking
- Fix frontend timezone drift in subscribed_at date picker
- Fix provider deletion index shift for expandedProviders state

frontend/src/api/admin/settings.ts

@@ -497,9 +497,8 @@ export interface WebSearchProviderConfig {
   type: 'brave' | 'tavily'
   api_key: string
   api_key_configured: boolean
-  priority: number
   quota_limit: number
-  quota_refresh_interval: 'daily' | 'weekly' | 'monthly'
+  subscribed_at: number | null
   quota_used?: number
   proxy_id: number | null
   expires_at: number | null
@@ -510,6 +509,12 @@ export interface WebSearchEmulationConfig {
   providers: WebSearchProviderConfig[]
 }
 
+export interface WebSearchTestResult {
+  provider: string
+  results: { url: string; title: string; snippet: string; page_age?: string }[]
+  query: string
+}
+
 export async function getWebSearchEmulationConfig(): Promise<WebSearchEmulationConfig> {
   const { data } = await apiClient.get<WebSearchEmulationConfig>(
     '/admin/settings/web-search-emulation'
@@ -527,6 +532,16 @@ export async function updateWebSearchEmulationConfig(
   return data
 }
 
+export async function testWebSearchEmulation(
+  query: string
+): Promise<WebSearchTestResult> {
+  const { data } = await apiClient.post<WebSearchTestResult>(
+    '/admin/settings/web-search-emulation/test',
+    { query }
+  )
+  return data
+}
+
 export const settingsAPI = {
   getSettings,
   updateSettings,
@@ -544,7 +559,8 @@ export const settingsAPI = {
   getBetaPolicySettings,
   updateBetaPolicySettings,
   getWebSearchEmulationConfig,
-  updateWebSearchEmulationConfig
+  updateWebSearchEmulationConfig,
+  testWebSearchEmulation
 }
 
 export default settingsAPI