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

Just like other companies that have similar low alternative services. They let you do it. They log and flag and as soon as you have a problem, instead of working to solve it, they see you are on the flagged list and you are now banned from the service. People who find themselves banned from an airline when they miss a connection or report lost luggage have done some other activity that got them to be flagged as a potential unwanted customer.