fix(dashboard): let the web-search route model be cleared again

[MikeTuev]

Problem

settings.webSearchRouteModel (layer 2 of #4481) reroutes every request carrying a native web_search tool to the configured model — open-sse/services/webSearchRouting.ts::resolveWebSearchRouteOverride. Its own help text says "Leave blank to disable".

But the dashboard field renders ModelSelectField without allowEmpty:

// src/app/(dashboard)/dashboard/settings/components/RoutingTab.tsx
<ModelSelectField
  value={String(settings.webSearchRouteModel ?? "")}
  onChange={(v) => updateSetting({ webSearchRouteModel: v })}
  placeholder={t("webSearchRoutePlaceholder")}
  disabled={loading}
  ariaLabel={t("webSearchRouteTitle")}
/>

ModelSelectField forwards placeholderDisabled={!allowEmpty} to <Select>, so the blank option is rendered disabled. Once any model is selected, the operator can never go back to blank from the UI — the override stays pinned forever.

Observed live on a Claude Code + Anthropic-provider setup: web search silently stopped working because every search request was being rerouted away from the configured provider.

{"tag":"WEBSEARCH-ROUTE","msg":"web_search tool → model override: claude-opus-5-5 → cx/gpt-6-astra"}

The only recovery was editing the settings row in SQLite by hand.

Fix

Pass allowEmpty (blank option becomes selectable) and allowCustomInput (free-text field, so an unlisted/self-hosted target can be typed and blanked) — the same combination the Modality Bridge Vision/Audio pickers already use with this component.

No API change needed: settingsSchemas.ts already types it as z.string().max(200).optional(), and the hidePaidModels guard in src/app/api/settings/route.ts only runs if (typeof body.webSearchRouteModel === "string" && body.webSearchRouteModel.trim() !== "").

Validation (TDD)

New tests/unit/ui/web-search-route-model-clearable.test.tsx (vitest + jsdom, 3 cases):

1. with allowEmpty the picker renders a non-disabled blank option and selecting it emits onChange("");
2. without allowEmpty the blank option is disabled — pins the old behavior so the regression is unambiguous;
3. source guard: RoutingTab must keep passing allowEmpty to the webSearchRouteModel picker.

• Before the fix: case 3 fails (expected false to be true).
• After the fix: 3/3 pass.

[MikeTuev]

⚠️ base-red inherited: #14547 — the two red checks are not from this PR (it touches exactly 2 files: RoutingTab.tsx + a new test).

Docs Gates (fast-path) → step Docs accuracy (fabricated-docs + i18n mirrors, strict). Reproduced on the base:

✗ In code but missing from .env.example: 1
   - DEEP_HEALTH_CHECK_ENABLED

DEEP_HEALTH_CHECK_ENABLED is read at src/app/api/monitoring/health/route.ts:46 and was introduced by 7d5292fcac — fix(health): opt-in deep check sampling completions surface (#14236) — without the matching .env.example / docs/reference/ENVIRONMENT.md entries.

API Route Typecheck → 4 new diagnostics, all in files this PR does not touch:

✗ open-sse/executors/auggie.ts TS2769 (baseline 0, live 4)
✗ open-sse/executors/auggie.ts TS18047 (baseline 0, live 4)
✗ src/app/api/v1/combos/projectCombo.ts TS2459 (baseline 0, live 2)
✗ src/app/api/v1/combos/projectCombo.ts TS2724 (baseline 0, live 1)

Both belong in their originating branches, not here.

[diegosouzapw]

Clean, well-scoped fix — traced placeholderDisabled={!allowEmpty} down to the <Select>
and confirmed the blank option really was unreachable once a model had been picked. Ran
your new web-search-route-model-clearable.test.tsx at HEAD: 3/3 pass, and it exercises
the real ModelSelectField component, not a reimplementation. Merge-ready.