This is really important: Health care should be really, really easy to afford.
For some comparison: Since I moved out of the states, I pay less than $300 a year out of pocket (in general). My biggest expenses - a neurologist, for example - costs less than $50. As does the ER, honestly, or the clinic that deals with injuries (the ER will send folks over there). The rest is taxes, and taxes are lower than federal + state + insurance premiums in the US.
Additionally, it isn't just insurance. If you work while the normal doctor is open and you don't have paid sick days, you might not be able to go to the doctor even if health care is free simply because you can't afford to miss work. You need paid time off for this sort of thing - and honestly, parents need more days since they have responsibility for than one person.
Ironically they tend to be the uninsured because the ER can't turn you away. The solution is to get them insured.