I think you're missing what the parent is saying: since the majority are poor, they plainly don't rule (since then they wouldn't remain poor). If democracy is defined as "majority rule," and we plainly see that the majority in the US do NOT rule, then the US is not a democracy.

See Gramsci's work on hegemony. Or (slightly easier) Tressel's "Ragged Trousered Philanthropists" which makes exactly the point about how the poor become convinced that it's better to give more things to the rich.

They aren't is the point. They would be if we lived in a democracy, and according to OP we don't.

If Party D promises free money to the poor and Party R does not, do you expect no poor people to vote for Party R? What if Party R has other things about it which appeal to the poor?

The poor have the opportunity to vote, and rule indirectly, and yet for many reasons often vote against their own best economic interests. Some of those reasons are good, and some are not.