I just can't see why so many people think of her in an American context when she knew fuck-all about the west.
America is not just the West. After all, there's a lot of people in America who know a lot about America, but very little about the rest of the West. If you look at Atlas Shrugged, it's quite clear that everything is deliberately vague beyond the borders of the US, which might be her ignorance of Europe and South America or that she was writing for a mostly US audience (who would not be familiar with those places either).
But even granting that, she had lived in the US for over a decade before the publication of Anthem, her first libertarian-style fiction and over three decades by the time of the publication of Atlas Shrugged.
If Stalin had parachuted a writer into the USA with instructions on writing something to undermine democracy it wouldn't have been as effective as the damage that Rand did with her rants about anything that wasn't aristocracy.
And I'm sure, if I had parachuted into Mother Russia with instructions to undermine the regime with my knowledge of orgami, that would have been super-effective too.
It's "twilight" for people who think they were born to rule.
You might find this hard to believe, but people who think they're born to rule, don't need a book to justify their world view. Rand was a bit crazy, particularly in her later years, but it's ridiculous to assert she was undermining democracy with words (actually it would be ridiculous to assert that she was even trying to - she does have a long pro-democracy history, let us note). It is impossible to do that just as it would be impossible to overwhelmhe USSR with a few small bits of folded paper.
What I believe happened here and why Rand was so influential was because she was a voice for groups and ideas which had long been neglected and derided. I speak not just of the imaginary people who supposedly thought they were born to rule, but somehow couldn't avail themselves of the multitude of utopian, authoritarian belief systems tailored to them, but also the people who grew tired of the ever increasing dysfunctionality, demands, and hypocrisy placed on them by society and authority.
And seriously, how is Rand supposed to have undermined democracy anyway? I note that we in the US currently have a president who is the antithesis of anything Rand believed in, who is casually slipping the US into another Vietnam-style war in Iraq, created the most destructive social program (the Obamacare thing) in forty years, has a variety of 0-9 losses at the Supreme Court from breathtakingly irrational, unconstitutional actions and arguments (typical looter logic in Atlas Shrug but without the compliant judicial branch to make it work), and a variety of creeping acts of tyranny.
Meanwhile we have a senator, named after Ayn Rand herself, who just opposed renewal of the Patriot Act, which is probably the single biggest step toward US tyranny in the last half century, by filibustering it at a crucial period prior to its renewal.
I suspect you might not have a clue what undermining a democracy entails. It's not words that undermine a democracy, but the crushing of freedom to act, to speak, to think, which undermines democracy.