The history of national highways goes way back before the Federal Highway Aid Act of 1956 that established the U.S. interstate highway system. The first national road was legislated into existence all the way back in the early years of the 19th century.
They were pikers back then...
“Was that really the primary purpose of the interstate highway system, or did they just say that in order to justify spending Federal money on it?”
No, it was really the purpose. Eisenhower was faced with a choice of developing rail transport or creating the interstate system. Based on his observations during WWII, he chose the autobahn.
It’s much easier and quicker to bring a stretch of bombed-out road back into service than it is a stretch of bombed-out rail.