[Demo][Copilot App] Add product search endpoint and UI filter

## Demo goal
Show issue-to-PR autonomous implementation in Copilot App.

## Requested change
- Add `GET /api/products/search?q=`
- Add a search input in `demo-app/public/index.html` and wire it in `demo-app/public/app.js`
- Keep behavior backward compatible with existing `/api/products`

## Presenter cue
Assign this issue to Copilot in the Copilot App and let it produce a PR from this issue.

## Files
- `demo-app/server.js`
- `demo-app/public/index.html`
- `demo-app/public/app.js`

Start with demo-app/server.js to understand the existing /api/products behavior, then inspect demo-app/public/index.html and demo-app/public/app.js for the current product display. Add the search endpoint and connect the input filter while preserving the existing endpoint behavior. Done means searching by q updates the UI and the original product listing still works.

Written by the indexing model from the issue text.