which they also pay a vanishingly small portion of.

They pay as much as the billionaire when buying something - sales tax, property tax, employer's health care etc.
Because all of that is included in the price.

Of course it is different from country to country, but genereally the poor pay just a little less % of their income as taxes than the middle class. And sometimes the ultra-rich pay even less (even with higher income tax) because they don't buy as much (% of income). Poor buy with all their money just for daily stuff.

we simply run a deficit.

Which is used as a reason to cut expenses for the poor.

Isn't that exactly what happens with subsidized health, education and housing?

No. I am surprised you still haven't heard that US health is far more expensive than in all the states where everyone (=subisdezed) has access. Or free education. And if I read about some housing prices in the US...

reinvest it in something else?

They put that money somehwere to make more money.

Don't confuse investment with speculation! Investment is production, speculation is money, to put it simple.

Every stock market transaction after the initial stock offering is speculation!

It's true, unlike other places America has no fixed castes or classes, anyone can become rich or poor, someone who is rich today could be poor tomorrow or vice versa.

That is true even for North Korea.
The American Dream is just that - a dream.
In reality social mobility in the US is far more unlikely than in most european countries - and the most "agile" ones are the "socialist" North European countries.

Hm... I could not find the article I wanted to show you.
But here is an OECD pdf: Page 178
http://www.oecd.org/centrodemexico/medios/44582910.pdf
Earnings of sons compared to fathers: The USA is one of the worst countries for correlation. Even my already very bad Germany is 1/3 better.

And if you look at this graph (Wikipedia)

you see that the US is not only bad for intergenerational chanes, but also for general inequality.