[PeterisP]

"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.