> How do they plan on stopping me from using Claude code with the Claude in chrome plugin, or some other computer use agent?

They don’t.

When a dispute that either party(you or United airlines in this theoretical case) decides to litigate an argument far enough that it becomes clear you used an AI agent like Claude then you are going to run afoul of the CFAA[1] and are going to get fucked legally because they stated who could access their site and you violated it with your agents against their stated will.

This is a legal ward, not a technical one. Violate it at your own peril.

[1] https://en.wikipedia.org/wiki/Computer_Fraud_and_Abuse_Act