docs(vercel-ai): use Vercel AI SDK’s supported integration setup

[shrey150]

The Vercel AI SDK integration already has a supported host tool API: createMCPClient. Lead the public guide with that integration route and a portable configuration for an existing application, instead of requiring a full Stagehand checkout.

This is an SDK/MCP integration, not a separate marketplace plugin. The source example remains available.

Stacked on #2971 for the shared @browserbasehq/stagehand-mcp package. The first 0.1.0 release is required before the portable npm command is available; the guide explicitly identifies this prerequisite.

E2E Test Matrix

Command / flow	Observed output	Confidence / sufficiency
Host-specific example/config check	TypeScript example typecheck passed; real AI SDK client discovered run, snapshot, and screenshot from the locally packed server and closed.	Host contract and configuration; no full model-loop benchmark claimed
Shared packed MCP server from #2971	Real Browserbase navigation, persistent snapshot, JPEG, and shutdown passed on example.com	Shared browser implementation; npm download awaits publication

Documentation validation: a combined snapshot of all nine updated integration guides passed mint validate and mint broken-links --check-anchors --check-redirects --check-snippets.

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[cubic-dev-ai]

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