I think it's clear that the basic biological differences between men and women extend well beyond just our genitals. Certain behaviors and characteristics can certainly describe women as well as men.

I understand what you are trying to say, but it is really important you are specific about these things. There are a lot of misconceptions about this, and in the worst cases it's being used to defend sexism.
Furthermore, in reality all of our behavior is modified by our environment to some extent. It's so conflated we really aren't sure how much biology plays a role and how much is just environmental factors.

I could sum it up by saying that there are very little difference between male and female brains. And that there is almost nothing that men or women couldn't do that the other sex/gender can.

Moreover, variance between groups is very large. If you for instance take the idea that testosterone makes men less likely to cry, you might say that crying is therefore more feminine. However within the group of men, men vary a lot with regards to their levels of testosterone. Even more so, so do women. A man with very little testosterone might then actually have lower levels than a woman, and be more likely to cry. This however doesn't make this person any less of a man.
Some women have higher muscle mass than some guys. This doesn't make those guys less manly.

This is important to make explicit.

You refer to another comment you made, I assume it is this one:

I suppose this can be said of anything, though, couldn't it? Of course women can be leaders, builders, protectors, intellectuals, etc too. But virtually all societies do and have chosen to assign certain characteristics as more associated with women or men, and these can be positive, constructive and aspirational points of pride for each.
I guess I just don't see the harm in emphasizing and celebrating the good stuff that can stereotypically characterize either gender, especially in lieu of all the bad stuff.

This is an important point. When somebody says: "Y is for men, and X is for women", they implicitly also say "X is not for men, and Y is not for women". To pride men on being leaders also holds an implicit assumption that women aren't really leaders. Because if women would be, men wouldn't specifically be praised for it because it would be a genderless thing.

And that also forms the danger of prescribing various behaviors or jobs to a gender. To say something is for men, makes it automatically less so for women. And vice versa.

This is where my first point comes in again. There are actually very few things that men or women cannot do. For instance, research found no difference between male or female leadership effectiveness. If anything, women might actually be slightly better leaders because they engage more in transformational leadership.
So then why should leadership remain a point of pride for just men?

If there are barely things that any gender cannot do, why do we insist on encouraging some genders to do it, but not the other one?
Wouldn't it be far better to just encourage a person to do it regardless of gender? Their success is determined by their personality and determination, not what's between their legs (or what they identify as).

The harm you speak of is there, it's kind of hidden though, but it's there.
For instance the idea that women are the caretakers has spread throughout our society. And now when men want to take up that role, they are faced with weird reactions, or even ridicule. Male nurses can get ridiculed, female scientists might be constantly questioned by their environment "why don't you choose x instead?".
Even worse, it causes implicit biases towards genders. For instance, some studies find that women are less likely to be promoted (example).

I guess as long as it doesn't affect you it might not seem like a big deal. But if you're a guy that really wants to be a nurse, or a girl that aspires to work at NASA, facing those social norms at every step of the way can be a real pain in the ass.
If you're a women who gets discriminated against in the workplace, or if you're a guy that gets ridiculed for being emotional, these things get harmful.

There are also other gender norms that have harmful effects. Men being taught not to cry is an important one. Even if men cry less, and tend to prefer other reactions, it is still incredibly harmful to teach them not to cry. Crying is an important reaction of the body to relieve stress and emotion. Furthermore men aren't taught to work with their emotions, instead they are taught to just put them away. Psychologically, this is one the worst things you can do with your emotions.
So some ideas what it means to be a man (or a woman) simply just need to go for the sake of our mental health. It's not necessarily about forcing men to do this. It's about stopping to teach them NOT to do it, to ridicule them for it. So that they may have the option to if they need it.
I've said this in a similar way in the past; we as humans are born with various coping strategies. But then we decide that women get some of them, and men some other ones. Why? We should all be able to have every coping mechanism available to us as humans. If you want to punch it out on a punching bag, do it. If you want to cry it out, do it. If you want to write about it, do it. If you want to talk about it with your friends, do it.