Agent UX: go from trying a Gradio UI to calling it from an agent in one click

Needs designing. Make it much easier to go from "I'm trying out this Gradio UI" to "I want my agent to run this", since we think this flow will drive the most growth.

Idea

• After a user runs a prediction, the footer briefly highlights
• Clicking it shows the exact request the user just made, as an API call (Python/JS client, cURL) and as an MCP/agent tool call, ready to copy
• Could build on the existing API recorder in "View API"

Related: #13915 (footer redesign)