This only works in a country where everyone is an upper-middle class 6-figure earner who can actually afford to pay "every day" medical costs out of pocket, plus the cost of a catastrophic care policy. Have you checked the bill statements your doctor sends and your insurance pays? Most Americans can't afford to pay out of pocket for even basic preventive care, or pay for their prescriptions for chronic illness. That's what happens when wages stay flat for 30 years and health costs rise at a rate far exceeding inflation. The result is that insurance becomes not only catastrophic care, but a "pay as you go" system for basic care as well.
Further, insurance coverage of preventive care wasn't invented by "greedy politicians". It was invented by insurance companies. It is in the financial interest of insurance companies to make sure everyone gets preventive care, so as to decrease the payouts for catastrophic illness later. Long before Obamacare, insurance companies got the idea on their own to cover preventive care. That includes contraception, since pregnancy is expensive.