How can the nicest men in the conditioned subset be as nice as the average person in the whole population? I don't grasp the bolded sentence beneath from Jordan Ellenberg's article. His diagram no longer renders, so I use these. 

How can the nicest men in the green triangle be as nice as the whole population's average man? Wouldn't they be NICER than the whole population's average person?  $\color{#c033ff}{\text{Average niceness in green triangle} =\dfrac{0.4 + 1}{2} = 0.7.}$
But $\color{#a96f63}{\text{average niceness in population} =\dfrac{0 + 1}{2} = 0.5.}$ 
This outcome is intuitive, because the green triangle excludes the louts with $0 \le niceness \le 0.4$. 

Now the source of the phenomenon is clear. The handsomest men in your triangle, over on the far right, run the gamut of personalities, from kindest to (almost) cruelest. On average, they are about as nice as the average person in the whole population, which, let’s face it, is not that nice. And by the same token, the nicest men are only averagely handsome. The ugly guys you like, though—they make up a tiny corner of the triangle, and they are pretty darn nice. They have to be, or they wouldn’t be visible to you at all. The negative correlation between looks and personality in your dating pool is absolutely real. But the relation isn’t causal. If you try to improve your boyfriend’s complexion by training him to act mean, you’ve fallen victim to Berkson’s fallacy. 

 A: In the sketch presented in that article, the bolded sentence is not really true.  (There's an excerpt from Jordan Ellenberg's book here that covers similar ground as the Slate article, and that still includes the sketch as of this writing.)
In that sketch, the long side of the 'triangle of acceptibility' goes from about (1/3,1) to (1,1/3).  So the mean niceness of the handsomest men (those with handsomeness=1) in the triangle would be approximately 2/3, whereas the average niceness of all men is of course 1/2.
But if you modify the sketch so that the long side of the triangle goes from (0,1) to (1,0), like this:

then the acceptable-and-very-handsome men will be the quadrilateral shown in the figure.  And so their average niceness will be approximately 1/2.  And if you really restrict yourself to the handsomest men (those with handsomeness=1, and equivalent to the case where the quadrilateral becomes infinitely thin), their average niceness is exactly 1/2, the same as the average niceness of all men.
And drawing the sketch this way still provides an effective illustration of Berkson's fallacy, because it still retains the negative correlation between niceness and handsomeness within the population of acceptable men.  Maybe the author had an earlier version of the sketch that looked more like my sketch, and neglected to update that one line of the text after they updated the sketch?  (This last is pure speculation on my part...)
A: 
How can the nicest men in the conditioned subset be as nice as the average person in the whole population?

That was not the situation sketched in the text.

The handsomest men in your triangle.... on average, they are about as nice as the average person in the whole population, which, let’s face it, is not that nice. 

So you mean: How can the most handsome men in the conditioned subset be as nice as the average person in the whole population?

That altered version is answered by Adam L. Taylor (for a particular triangle it is true)
But aside from that, the article actually tries to explain something different and did, I believe, not phrase it well. The paradox is that in the selected population you observe a correlation between handsomeness and niceness. (at least that is what I believe it was supposed to explain because the title reads "Berkson's fallacy: Why Are Handsome Men Such Jerks?")
The most/more handsome men inside the triangle are on average less nice than the average of all men in the triangle.
The Berkson's fallacy relates to the point that this correlation is due to the selection (the triangle, the selection based on a collider variable) and that the correlation is not true for the entire population (square), or not (necessarily) a causal relation (and it is fallacious to consider it like that).
See in the example below where a linear model is fitted to the data. You can see on the right that the selection (triangle) makes the relationship negative. The more handsome the men are, the less nice they are.


links to the original images that did not render


*

*The great square of men:
https://slate.com/_components/image/instances/cq-article-87d2464659bfec0587c3608d700371ec-component-7@published
https://compote.slate.com/images/e7b77786-4cd3-499e-ac97-6fdd73939ab3.jpg

*Triangle of acceptable:
https://slate.com/_components/image/instances/cq-article-87d2464659bfec0587c3608d700371ec-component-10@published
https://compote.slate.com/images/689ce72d-44f5-4e1d-9c36-6132e8d153b1.jpg
