To answer this, I have to let you in on philosophy's dirty little secret: Almost all of its big names commit logical fallacies by the truckload. Try reading Descartes' Meditations. The clarity of his writing makes it easy to see the flabbiness of his arguments, but he's far from the worst offender.
Once you digest this hard truth, it is only natural to wonder: Why do smart philosophers make silly arguments? This deep question was best answered by the 18th-century philosopher Thomas Reid. Long story short: Philosophers are embarrassed to just say "It's obvious," so they hide behind convoluted thirty-step arguments that they construct after the fact to rationalize their intuitions. As Reid puts it:
[W]hen we attempt to prove, by direct argument, what is really self-evident, the reasoning will always be inconclusive; for it will either take for granted the thing to be proved, or something not more evident; and so, instead of giving strength to the conclusion, will rather tempt those to doubt of it who never did so before.
Of course, once some philosophers start trying to prove the obvious, others make even more bizarre efforts to prove the absurd.
Admittedly, a philosopher can be brilliant and creative without appreciating any of the above. But for a philosopher to be wise, to have informative things to say about the Big Questions, it's essential.
Ayn Rand is one of the wise philosophers. Despite some low-quality arguments, she largely sticks to promoting obvious truths, and ridiculing others' absurdities. The payoff: Genuine progress on the Big Questions.
When Rand wrote her major works, movements like logical positivism, Marxism, and Existentialism were all the rage, and obvious truths were out of fashion. Obvious truths like: There is a real world out there, and old-fashioned bourgeois reasoning is the best way to understand it. Her rejection of the contrary view is sharp but not shallow:
"We know that we know nothing,' they chatter, blanking out the fact that they are claiming knowledge–"There are no absolutes," they chatter, blanking out the fact that they are uttering an absolute–'You cannot prove that you exist or that you're conscious,' they chatter, blanking out the fact that proof presupposes existence, consciousness and a complex chain of knowledge: the existence of something to know, of a consciousness able to know it, and of a knowledge that has learned to distinguish between such concepts as the proved and the unproved.
When Rand moves from epistemology to ethics, she adds originality to her common sense. Many ethical teachers - whether Jesus or J.S. Mill - enjoin you to love your fellow man. Rand, like the child in "The Emperor's New Clothes," essentially retorts "Have you met these people?" Love everybody? Most people simply don't deserve our love. Even an extremely likable stranger would be out of line to demand your help, and most of your fellow men fall well below that threshold. Furthermore, harping on the creepy duty to love everyone makes it easier to forget our primary duty to our fellow men: to leave them alone.
These ethical truisms are the base for Rand's critique of the welfare state. Why exactly are we supposed to pay taxes to help the poor? We aren't legally obliged to help our siblings or friends, so why are we legally obliged to help perfect strangers? And in the process of forcing people to pretend they love poor people they never met, don't we breach our far more obvious duty to leave people alone?
These are not irrefutable proofs, but they are far more convincing than anything in Descartes' Meditations.