This could be an email, but of course the business doesn't want this as it would create a record of interaction. Whenever I had to use live chat it was a tactic to wear you down.
I just now chatted with Amazon, and not only do they not send (or ask to send) a transcript, they even clear the chat once the agent has left so you can't even save it yourself.
I can't imagine any other reason but "we don't want the customer to be able to have a record".