It has a big banner that says "Research preview: The browser extension is a beta feature with unique risks—stay alert and protect yourself from bad actors.", and it says "Join the research preview", and then takes you to a form with another warning, "Disclaimer: This is an experimental research preview feature which has several inherent risks. Before using Claude for Chrome, read our safety guide which covers risks, permission limitations, and privacy considerations."

I would also imagine that it warns you again when you run it for the first time.

I don't disagree with you given how uniquely important these security concerns are, but they seem to be doing at least an okay job at warning people, hard to say without knowing how their in-app warnings look.