Of course we built a road system - without paved roads cars are mostly useless. We have build rail systems in the past as well. The problem isn't unsolvable by the nature of the problem. It is unsolvable because you can't get the right people to make long term investments in it (don't look to government - anyone who tries will be voted out eventually at best the system changes course from one okay system to a different incompatible system, though more likely they just stop investing).

They don't live a good distance from work because they want to they do because they can't afford to do otherwise. If you told one of the multitude of people who commute via car they have to spend 5000 on a clown car and an an hour to their commute which amounts to 500 hours annually or 20 days spent commuting they would be in your front yard with torches.

Instead what you need is enough housing to bring its price low enough for more people to live closer to work and enough transit that people want to use it not a clown car mandate.

When you have successfully convinced most people to commute differently you can start to limit cars not before.