That is true; however, in a majority of U.S. states, those damages would not be due from Avis (even if they own the car) but exclusively from the driver of the car. So, if you take one of their cars and kill someone, all they would be concerned with is, how badly did you dent the hood and when can they start renting it again.

There are a few exceptions to this (I believe New York is one of the exceptions, not sure of the others) which have what is known as "strict liability", in which the owner of a car can be held liable for damages caused by the car. However, in most states the law is not written this way and the owner of the car bears no risk for damages caused by the driver of the car. In those states, rental companies do generally include some minimal amount of liability protection in the base rate, precisely to protect themselves. In other states, they would not do so because there is no need.

That may be one reason they're willing to let you drive away without liability coverage.