The USA did such a startup. It’s called healthcare.gov, AKA Obamacare. It allows you to get health insurance, without consideration for pre-existing conditions and provides you with significant tax credits to offset the premiums, based on your income.

Most of the policies are high-deductible, which means you’ll end up paying for the “scratches and sniffles” yourself but have a per-year maximum-out-of-pocket that can be planned for.

It’s not perfect but it at least provides an option whereas before if you had any non-trivial pre-existing condition you couldn’t find any insurance at all.

Maybe it is wrong to ask insurances to insure people with pre-existing conditions. Perhaps such cases should be considered charity or social help, and convered by the state? Just wondering if reframing the problem as such could help. I'm not from the US, so not sure what exactly the issues are.

It seems healthcare.gov is not good enough yet? I don't think classifying government institutions as startups is really correct, though. They have infinite money supply and no requirement to be profitable.