Redesign ts-react-chat: clean up UI and merge in ts-react-media (fal) example

Background

The examples/ts-react-chat example has grown organically and the UX no longer reflects the SDK's structure well:

• routes/index.tsx is 642 lines — the home page IS the chat, mixing landing/navigation with the chat surface.
• routes/server-fn-chat.tsx is a 113-line near-duplicate that exists only to demo wiring useChat to a TanStack Start server function (added in #509). Same feature, different connection adapter.
• The sidebar lists every generation type as a separate route (generations.image, generations.speech, generations.audio, generations.transcription, generations.video, generations.summarize, generations.structured-output), making the example feel like a directory rather than a product.
• Meanwhile, examples/ts-react-media is a separate, cleaner example that demonstrates fal image/video generation with a tabbed UI — but a new user has to discover it on its own.

Proposal

Redesign ts-react-chat so it works as a single, cohesive demo app:

1. Separate home from chat

• Home page becomes a real landing screen with a prominent "Start chatting" entry point + cards/links for the other demos.
• Move chat to a dedicated /chat route.

2. Collapse the server-fn duplicate

• Delete routes/server-fn-chat.tsx.
• On the chat page, expose connection mode (HTTP API route vs. server function) as a toggle/setting so users can see both wirings in one place.

3. Merge in ts-react-media

• Pull the fal-powered image/video generators from examples/ts-react-media into ts-react-chat so a single example covers chat + media generation end-to-end.
• Keep the tabbed image/video UX from ts-react-media; consolidate with the existing generations.image / generations.video routes (the existing ones use OpenAI; the media example uses fal — show both as provider options).
• Once merged in, delete examples/ts-react-media.

4. Sidebar / IA cleanup

• Group generations under a single "Media" section with provider tabs rather than one route per provider/type.
• Reduce visual noise; the current sidebar has ~10 entries.

Out of scope

• No changes to the SDK itself — example-only refactor.
• Other framework examples (ts-solid-chat, ts-vue-chat, ts-svelte-chat) follow separately if this lands well.

Context

Discussed in #509 (server-functions PR) — defer the UX work so #509 stays focused on the SDK surface.