Their large incomes are made possible by infrastructure. The analogy doesn't make much sense to me since even the most prolific billionaire didn't build much of this infrastructure.
As a class they did. Some of it was built by billionaires past, but is maintained by the current generation of the well to do. Regardless, the middle income household certainly did not build it.
Oh, I didn't know that "billionaires as a class" digged roads, erected buildings, manned trains, stores, telephones, farms, factories... "The low income households didn't build it", you have a very twisted definition of what that means.