You're right that the poor pays low or zero in income-taxes. Nevertheless one of the major differences between different tax-systems, is to which degree they're progressive.
(Notice that I consider health-insurance and suchlike to be part of the tax-system. Otherwise it becomes impossible to compare and contrast different tax-systems. When we're talking wealth-distribution, it's fairly irrelevant if you've got 30% taxes that includes health-insurance, or if you've got 17% taxes plus 13% for health-insurance.)
Income-tax is progressive everywhere as far as I know (though with very different slopes), but one of the main differences between the rich and the middle-class is that the middle-class, overwhelmingly live from salaries on which income-tax is paid, while the primary income of the rich is capital gains of various sorts.
If one person has a salary of $100K, and another person has a salary of 0$, but a stock-portfolio that grew in value by $100K, then those two people will pay very different taxes - despite the fact that they've both made the same amount of money. (indeed, the latter guy can easily get away with paying -zero- in taxes in many jurisdictions)
Offcourse huge jumps in social class, especially upwards, are much more rare than smaller climbs. That's true everywhere. Nevertheless there's significant variation in social mobility between different nations, and USA ranks rather poorly for a wealthy nation. I guess I should be specific here, when I say social mobility, the actual proxy number that I'm mostly thinking of is: "To what degree does the income of your parents predict your future earnings?" i.e. if you guesstimate that children end up in the same relative place as their parents, how often are you wrong ? (being often wrong, would indicate high social mobility)
For example, according to the latest OECD-report, (available here: http://www.oecd.org/dataoecd/2/7/45002641.pdf ) the income of your father, explains 47% of the variation in your income. This compares to numbers under 20% in Scandinavia and low thirties for Spain and Germany. (but the UK is even less socially mobile with fathers income accounting for 50% of the variation)
Same for education: In USA your parental background explains 64% if the variation in PISA test-scores, whereas in Scandinavia it accounts for about 35% of the differences.
In practice, this means in USA it's comparatively rare for children with resourceful parents to do badly, and comparatively rare for children of marginal parents to do well. In Scandinavia and much of the rest of Europe, it's much *less* rare. (allthough those with resourceful parents have an advantage everywhere, the advantage is smaller)
There's many reasons for this. I *do* think additional government-provided for-everyone services is one piece of the puzzle, but offcourse that's just one small piece of it, and there's many other reasons. (for example, making access to education independent of parental income obviously results in increased mobility - saving for your kids college is a non-topic in much of Europe)