Even if the constitution can be handwaved, a Russian woman would never be elected during the Cold War, regardless if her views were "sane" or "insane."
Having read a very detailed biography of her written by Anne Heller, it is clear that Rand had a lot of emotional baggage that was eating away at her, and the stress of a campaign could kill her, if she ever ran for anything.
(I happen to like Rand, though I find a lot of her views insane as well, so I'll withold judgement on the political aspect of her career.)