> so much as fallout from much more recent segregation policies.

The existence of slavery and the view that Black people could be treated as livestock are why we had segregation policies.

The existence of slavery fueled white anger towards the idea that the "livestock" were suddenly being allowed to go to the same places and use the same services as white people. The generations of people who owned slaves raised the generations of people that enforced segregation, and those generations raised the people today who enforce racial discrimination and oppose racial justice.

And like you said, this isn't far-back history, segregation in the US was recent. People have this flowery view that racism just went away and that slavery just suddenly stopped mattering. But the effects are still very much with us today. A few generations isn't enough to eliminate a culturally pervasive, harmful ideology.

But how, in a democracy founded on the notion of equality, did segregation become a policy? How did the people who passed those laws get it into their heads that it would be a useful thing to do?

I agree plenty of history happened in between, but your notion that they're unrelated is absurd. Nobody today learned English from Alexander Hamilton, but nonetheless we not only understand the Federalist Papers just fine, we live by many of the ideas he expressed. Culture is handed down.

If you'd like to better understand that, I'd suggest you read Kendi's "Stamped from the Beginning", who gives an intellectual history of America's racist ideas. I also think you're mistaken to lay the current troubles at the feet of formal segregation policies, which were only active in one part on the US. Maybe go read "Sundown Towns", which shows that all across America we had towns with de-facto segregation without formal policies. And it takes only a glance at modern maps showing race to see we haven't solved the problem.