You're right. The US isn't a democracy, it's predominantly an oligarchy/republic. And as matter of fact, no government is purely a democracy. Most are hybrids to varying degrees. The way I categorized the forms of government was just a way to show how things work. Nothing prevents it from being a spectrum.
In the US only 50% of the population votes. The church doesn't pay taxes but lobbies the gov via proxies. Rich people definitely have more rights than the others. Just because they fight in court doesn't mean it's not called violence. Rich people are the authority in the US. They decide what you watch, what you buy and where you can spend your money. Just because they work for corporations doesn't mean there isn't a hierarchy. Rich people own America and they behavior of it's citizens, because they shape the contexts in which you live. They also own the rest of the world via Hollywood.