Fix chat realtime status and mobile nav

[BunsDev]

Summary

• Fix stale realtime connection status in the Svelte shell by reporting websocket status explicitly into ChatApp.
• Treat mobile navigation as a proper drawer with backdrop close, Escape handling, aria-expanded, and inert main/thread content while open.
• Avoid GIF library filtering while the picker is closed and refresh the embedded web bundle.

Why

After the rebase, the sidebar could remain on ready / Reconnecting… even with an established websocket because legacy $: reactivity tracked the socket object, not state hidden behind the connection getter. The mobile nav also slid over the timeline without a clean modal boundary on small screens.

Validation

• pnpm fmt:check
• pnpm typecheck
• pnpm lint
• go test ./...
• pnpm test:e2e
• pnpm exec playwright test --grep "mobile navigation"
• Browser review at http://localhost:8080/app on desktop, 390px, and 320px viewports