Many skills have become useless over the centuries. Chances are future generations will no longer be able to drive, but it's not a problem. My granddad had skills which I no longer have, and which wouldn't improve my life one iota by having them, even though they were important in his time.

I do not see the problem with automated cars, driving people around who can no longer drive themselves.

Also, we see very little trouble trusting our lives to busdrivers, pilots, traindrivers, ship captains,... SO why not trust our lives to programmers? Especially when you can already now prove that they are safer.

The problem of the interaction between technology and psychology will also become less important, as more and more automatic drivers will be used. These automated cars will be able to communicate amongst themselves, and will avoid crashing into each other in the crash scenario your post started with.

I believe Elon Musk already stated that he expects people driven cars too become illegal in the far future, since they will be seen as much more dangerous than the automatically driven variant.