How the hell am I supposed to run a department without ever having a closed-door 1:1 meeting with 1/4 of my leadership team? I ignored the warning and things of course played out just fine, but it did give me some pause that people would think that simply having a meeting would be too risky to consider.
I wouldn't risk my career for a man, so why should I risk it for a woman? The rational thing seems to be playing it safe.
The whole situation is nuts
If you take the apparent maximally safe course of action at every turn in life, you probably won't achieve much. It's risky to change jobs, to break up with your current partner, to start dating someone, to have kids, to buy a house, to change cities, to pick a college/major, to pick an initial career, to drive in the rain, to play recreational sports, to travel to a different country, to eat sushi, and 100s of other times in any given year.
If you are willing to do something important (and work-related) for a man that you are not willing to do for a woman, IMO, you should not be in a position of leadership.
A lot of mentoring and networking happens on a 1:1 basis, and if men refuse to meet women 1:1 in situations where they would meet with other men, then women will lose out on career opportunities.