Given that every time they add a road, more cars travel them, I'm starting to think we haven't reached enough road density yet. I think we should add more.

I think your heart's in the right place here, and I don't mean to insult you or start a flamewar here - but, didn't urban planners discover induced demand 100 years ago? It's the nature of the beast that, whenever more roads are built instead of more alternative transportation infrastructure, more people will choose to drive on those roads. Of course, these new cars aren't coming out of nowhere - but if you're looking for a house or a job, you're likely to pick one at least somewhat based on the commute. New residents and people looking for new jobs will look for ones they can drive to, which means more houses and jobs on the highways, which means more cars on said highways, which means more highways, ad infinitum.

Failing the highways (as is often true in places like Europe or Japan), people will instead plan their commute based on where they can go with the transportation options available to them - and you can transport more people per hour on a railway than on a highway by orders of magnitude, so while the number of people being transported is going to be the same, the space and infrastructure cost required to transport them is lower with most alternative methods (though I can't say if bike lanes or other specific infrastructure options besides rail are among these). The American rail infrastructure sucks right now, but we spend a LOT of money on highways, and if a significant fraction of that were allocated to railways, it seems very feasible that they could become practical for interurban use again, as they were in days gone by.