That's a very good question, lil. To me, there are two defining characteristics of presidentiality that should transcend gender. One is the ability to take in a lot of information (some contradictory) and a lot of opinions, and maintain an ability to separate the signal from the noise to make a decision about a course of action. Two is the ability to get others on board with one's decision making, which is primarily dependent on one's ability to communicate and connect with people. These two together, for me, are what define leadership generally, and presidentiality in particular. I think whether a woman or a man is at the helm of a government or a company or a group of any sort, formal or informal, that both of those traits define leadership.

Edit: I think that "decency", "morality", "integrity", etc, are all proxies for trust, which just makes it more likely to be able to connect with followers and adversaries alike, but they're not great leadership qualities per se. Trump is very good at connecting with followers, almost magically so. He is not good at connecting with adversaries, because no one trusts him.