As long as your roads are saturated, they have less throughput, not more. It is kind of like a clog in your toilet: more things are there in your pipes, but not much is getting through.
Fluid analogies are crappy because fluids flow more when you add pressure and traffic doesn't.
Well, traffic can flow when pressure is added, just not in a way that is very safe.
The reason "just shutting down all roads" doesn't make sense is that it doesn't solve the actual problem, which is that people want to both work in places with good jobs (traditionally often dense urban centers) while living in cheaper places that are far away and not spend significant chunks of their lives stuck in traffic. Shutting down all the roads only "solves" the traffic problem in a deliberately ridiculous sense (same as "just kill all humans").
I work for a big central business district employer. You can pay $150-250 a month to park or $75/week to take a motor coach bus from your suburban town. Those numbers drive behavior, and make for a better solution as folks who need flexibility can pay for it.