There are such areas in every country. USA might have more of them but not so much more that it cancels out the low population density entirely. You still have to have a road going through these areas.
A substantial fraction of the US's land area is locked up in Alaska, where there isn't "a road going through [the low-density] areas." What little long-distance infrastructure exists there is almost entirely driven by the existence of extractive industries (notably, but not exclusively, oil) that are lucrative enough to put in that infrastructure.
Rural Europe tends to be as lightly populated as, say, rural eastern US, not rural High Plains, let alone rural Alaska.
That road comes out of state or federal taxes though. It's still a coat but not relevant to the math about city budgets. And yes, there is so much more space in the US even without leaving populated areas.
The lower density in urban areas is still real though.
North of UK is almost empty. North-Eastern Germany has population density similar to USA which is pretty low compared to the rest of the country. France and Spain especially are sparsely populated outside of the big metro areas.
