Americans think they're "free and democratic" in the same way that aristocrats think they're better than everyone - it's been inculcated in them since birth, by every aspect of the culture and their upbringing, and as an axiomatic belief, it's not something they challenge.
There's nuance, of course, with people who are worse off in America seeing some cracks in it, but that's how you get the idiom about "Americans vote like temporarily inconvenienced millionaires" - they are so convinced the game isn't rigged against them that they vote assuming they will win at the casino one day.
