If women did not exist, birth control would not be necessary. If men did not exist birth control would not be necessary. I'm tabling the STD side of the equation, that's not a significant burden on the healthcare system AFAIK. The health burdens of the lack of birth control should be carried equally by men and women, since both are a requirement to cause the childbirth and produce the child.

If women did not exist, car insurance would still be a thing. If men did not exist, car insurance would still be a thing.

Car insurance and health insurance are not apples to apples when it comes to where the burden lies. They share some similarities, but are not the same.