I understand that American regulation is company friendly so you can essentially promise and do whatever.

However, did Tesla not also sell in other areas, like in the EU? Were those claims absent?

Because if not, the case would be seem to be easy: Tesla sells a car with self-driving capable hardware (advertised in a manner where your purchasing decisions would be influenced by that claim), has obviously not delivered and has therefore up to three attempts to fix the issue during whatever the warranty period is. If that doesn't happen, you, the customer, can simply invoice your money back. It doesn't seem like this would turn out well for Tesla.

I mean, I have done stuff like this before - not on a scale of a car, but pretty expensive electronics. The company in question (Samsung) pretty quickly decided to play ball. It gets difficult to prove after six month if the issue is a defect. But if a thing is simply missing or doesn't work as advertised, then the chances for the company to wiggle out of it are rather slim.