Remove ext-apps SDK-v1 shims when the package ships a v2 peer (ext-apps#702)

[cliffhall]

Follow-up from the #1633 Extensions stack (Phases #1738 / #1739 / #1740). Tracks removing the throwaway shims the Inspector carries because @modelcontextprotocol/ext-apps still peers on TS SDK v1 while the Inspector runs SDK v2 (2.0.0-beta.x).

Trigger

Close this when @modelcontextprotocol/ext-apps publishes a release whose peer is SDK v2 (upstream ext-apps#702). Installed today: @modelcontextprotocol/ext-apps@1.7.4 (peers @modelcontextprotocol/sdk: ^1.29.0).

Shim sites to remove (verified 2026-07-22)

Each is tagged in-source with TODO: drop when ext-apps#702 ships a v2 peer. This should be closeable by a single sweep:

1. core/mcp/apps.ts:27 — tool as Parameters<typeof getToolUiResourceUri>[0] (v1 Tool vs v2 Tool at the getToolUiResourceUri boundary).
2. core/mcp/apps.ts:78 — the UiMetaShape structural interface standing in for ext-apps' McpUiToolMeta / McpUiResourceMeta, which aren't importable under NodeNext because of an extensionless re-export in the package's .d.ts. Switch to the named types once upstream fixes that re-export.
3. clients/web/src/components/elements/AppRenderer/createAppBridgeFactory.ts:~218 — client as unknown as ConstructorParameters<typeof AppBridge>[0] and the Implementation cast (ext-apps' AppBridge peers on SDK v1's Client/Implementation).
4. core/mcp/inspectorClient.ts:~1698 (getAppRendererClient() Proxy) — translates ext-apps' v1 schema-first setNotificationHandler(schema, handler) into SDK v2's method-string form.

NOT added by Phase 3 (#1740)

The io.modelcontextprotocol/ui advertisement added in #1740 needed no ext-apps cast — ClientCapabilities.extensions is typed { [key: string]: object }, so { mimeTypes: [...] } is assignable directly. The extension id / MIME-type constants are mirrored locally in core/mcp/extensions.ts (UI_EXTENSION_KEY / MCP_APP_MIME_TYPE) rather than imported, for the same NodeNext//server-subpath reasons — those can optionally switch to importing ext-apps' EXTENSION_ID / RESOURCE_MIME_TYPE once the v2-peer release lands, but they are stable spec strings so it's low priority.

Acceptance

• Bump @modelcontextprotocol/ext-apps to the v2-peer release.
• Remove shims 1–4 above; replace UiMetaShape with the named ext-apps types.
• Optionally source UI_EXTENSION_KEY / MCP_APP_MIME_TYPE from ext-apps.
• npm run ci green.