our respective lots in life aren’t some arbitrary “lottery”

Its still a lottery, because you yourself cannot effect a single one of those things. You can only start to change or fix them once you are a decent ways into your life.

Sure we can trace every single little bit down to who and what it to blame for it happening, but that doesn't change the fact that you popped out and were dealt a hand that is either extremely lucky or terribly awfully not.

Why should the child of a crack whore have the same advantages in life as the child of two hard working parents?

Generally, because children are innocent and shouldn't always be held to suffer infinitely the sins of their parents. If the situation of your birth has already determined your entire lot in life, then why should they ever work hard to begin with? You've already decided their value is worthless before they could even talk. Which just creates a resentful underbelly of losers who know they will never make it and turn to just taking it instead, through violence and crime.

There is a wide valley between "give crack babies a fighting chance to not end up a gang banger" and "give them free college and money and everything possible."