Providing a safety net might or might not encourage startups, but it's misleading to suggest that this idea is somehow in opposition to allowing economic inequality. To decrease economic inequality, you have to confiscate the wealth of the rich, not merely give money to the poor.
As long as you allow people to become billionaires, you'll have huge economic inequality, no matter what's happening at the other end of the scale.
I didn't get into this, but Brook argues that allowing people to become billionaires has led to increasing costs of living in major cities, pushing out the kinds of people who start startups in favor of trustafarians, i-bankers, corporate lawyers, and the other kinds of boring people that billionaires breed. He's more focused on stories than economic analysis, but the argument seems plausible.
They're not necessarily in opposition, but the only payment method that the universal health care supporters ever seem to come up with is "take it from the rich".
If you want to give more money to the poor, the only place you can get it is from the rich. But the amount you'd have to take to fund universal health insurance wouldn't put much of a dent in inequality. To get rid of inequality, you have to explicitly ban being rich, e.g. with tax rates that go asymptotic at a certain point.
The other thing to do is to look after the bottom X% in terms of absolute poverty - making sure that they have basic food, medical, and so on. You could still have a lot of inequality, but at least no one falls beyond a certain point.
