"A car is much safer on the street with a cage or a bar. I don't know where this wives tale came from. Put padding on the bars."

1. car is safer, but are the occupants?
2. Calling it a "wife's tale", doesn't make it one
3. Next time some one hits you on the head with an alum baseball bat, tell them to pad it first, and you be fine.
4. cages are installed with the understanding the driver will be tightly secured by a 5+ 3"? wide restraint system, AND have on a Snell approved helmet. The reduction in free movement caused by the space intrusion with a cage, and the fact that rollbars have an approx .875" radius, somewhat if not greatly smaller then almost any other interior surface of a non caged car that the driver can normally contact when belted in, is likely the source of this "wife's tale".
5. There are many circumstances having a proper cage adds safety for the driver in an accident, but striking ones unhelmeted head on a cage has huge negative consequences, padded or not.