fix(website): repair and redesign the docs adapter picker #2435

Job	Run time
CI scope	1m 37s
Website — lint / test / build	2m 33s
examples/ag-ui — e2e	4m 31s
Website — e2e	2m 37s
Cockpit — build all examples	0s
Cockpit — e2e dispatcher	0s
PostHog — dashboards-as-code drift check	0s
examples/chat — e2e (${{ matrix.shard }}/4)	0s
Cockpit — representative capability smoke	0s
Cockpit — build / test	0s
Cockpit — e2e (${{ matrix.cap.angular }})	0s
Angular ${{ matrix.angular }} — packaged consumer	0s
examples/chat — python smoke	0s
examples/chat — e2e	0s
Cockpit — deploy smoke dry-run	0s
Library — lint / test / build	0s
Canonical demo → Vercel	0s
Cockpit — e2e	0s
ag-ui demo → Vercel	0s
CI — required	2s
Deploy → Vercel	-1s
Production smoke	-1s
	11m 18s

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