The fact that there's substantial disagreement for and against these principles is an argument that whatever morality Ayn Rand is arguing for, that morality isn't universal, it would not be universally accepted.

I am pretty sure that we all agree that humans will never have a complete universal morality because we put so many things into the bucket of morality that could not be addressed by a universal morality. There are people who find it immoral marry between races so no logical system could ever resolve their specific belief. Logically though, we could propose a set of core axioms that everyone could agree on and then build up a system from the axioms. This is what Rand attempts to do so from her logical system you can at least define some moral arguments as universal.

That's a novel argument. I believe what Rand was ultimately concerned with was a morality that is rational and objective; the idea being that one only needs to rely on one's own reason in order to judge what is moral and what is not. As to universality of moral doctrines, if I was feeling malicious, I'd say that the Catholic church was there first (that's what 'Catholic' means, after all). Some rather remarkable moral notions have historically been 'universally accepted', as you say.

"the idea being that one only needs to rely on one's own reason in order to judge what is moral and what is not" - I would argue that this is fundamentally impossible. A commonly used (and IMHO generally accepted) counterargument to that is "Hume's guillotine" or is–ought dichotomy; you can't derive "ought" statements (i.e. morality) solely from logic reasoning and factual "is" statements.

Of course, you definitely can derive a full system of morality logically if you start from a few "ought" axioms - many proposed systems of morality are done this way, and likely Ayn Rand's proposal as well. But in such cases these axioms are the subjective and potentially questionable part, and we don't have an universal agreement on them. If your system has 99% of logical reasoning based on 1% intuitively assumed axioms, that 1% carries pretty much all the weight.

We know where many particular "moral axioms" lead, but that only leads to disagreement about them when seemingly reasonable moral axioms logically lead to various outcomes that intuitively seem unacceptable, so we don't have anything approaching consensus; for every proposal there is enough substantial critique that it doesn't seem suitable to be the One True Perfect morality - perhaps we should keep looking, but perhaps it's futile, there's no strong arguments yet to say that an universal morality must be possible in the first place.

I don't think universal morality means universally _accepted_ - I believe the idea is that it's universally _true_.

Truth is not truth due to everyone agreeing on it.