I can now precise define sympathy and empathy. It's mimicry vs. identification. Sympathy is feminine. Empathy is slightly masculine.
Sympathy is merely monkey-see monkey-do. Feeling what
they're feeling, regardless of the thoughts behind it. The sympathetic act out the emotions the way they themselves always act out the emotions. The empathetic could act out the emotion the way the person feeling it would act out that emotion, and sometimes even does so act.
Let's imagine a sad man. His empirical reaction: he throws himself into his work. It affects quality, but that aside sadness makes him highly productive. He realizes instinctively that the work has a distracting or numbing effect which increases ease while he waits for the involuntary chemicals to finish doing their thing.
The woman will react like she always reacts to sadness: crying and iced cream. Whether this sadness comes from within or from "feeling" someone else's sadness is not relevant. The reaction is identical. Unproductive; instead, expensive.
Lacking empathy entirely, she believes on the inside the man is crying and craving iced cream. He's sooo manly for resisting these urges, right girls? Though probably it's not healthy for him to "repress" so much... The only way to make her not believe this is for him not to show her his sadness; to genuinely repress. She's dragged around involuntarily by her reflexive grass monkey instincts. She couldn't fight them even if her soul wasn't literally identical to the pile of instincts in question and logically incapable of seeing any sort of dissonance with them.
(It doesn't help that he probably is crying on the inside. If he does burst, it makes the woman think her nonsense is accurate too.)
A properly empathetic man would realize that seeing sadness would make the woman feel sadness and therefore cry, and the man's feeling of sadness will make him throw himself into his work, assuming the empathetic man knows the sad man well and has met a woman once or twice before in his life.
It is only necessary for one gender to care overmuch about reproduction. As long as it's handled, you get a next generation. However, reproduction is kind of critical, so they're likely to become completely obsessed with it, to the exclusion of everything else. In other words Evolution doesn't trust women to do what's best for baby unless it a) forces them to act and b) forces them to emote in a baby-suitable way so the action is suited to taking care of the scion.
Everything else has to be handled by the other gender.
Therefore, everything else can be handled by the other gender.
You may note that it's not necessary for both genders to have empathy and self-control. One gender can handle all that for both, as long as the genders are generally or approximately paired up. The gift of empathy allows the empathetic gender to control their presentation and thereby control the other gender, especially if their opposite ruled by immutable instincts that can't learn to hack the game.
Indeed sympathy is so important for women they'll actively practice sympathy if they don't get exposed to a proper variety of feelings (sad youtube channels), in exactly the same way a man will lift something heavy for fun, or go running. Though note part of this is that sympathy grows more numb, rather than less, with practice; untempered sympathy will knock a woman on her ass if her life has been too easy.
P.S. By inspection, if everyone were identical sympathy would be identical to empathy. Maybe a third of narcissism and egalitarianism is butthurt about the narcissist's predictions being false instead of true; clearly the solution is to coerce the other into being the self so sympathy can be good enough. It burns especially bad if the other's reaction is wise and effective, and the narcissist's reaction is senseless and self-destructive. When someone is capable, the solipsist narcissist immediately concludes they themselves are capable; the narcissist handles it poorly when they try and fail.
Similarly if a culture is collectivist and domineering enough it can blur the difference. States, like ant hills and bee hives, prefer women, because the sympathetic are easy to manipulate. They have nice predictable reactions. Thus States try to breed and train their men into being women. In the short term, this even works out for them, and they get to feed their kratia addiction enough to outpace their growing tolerance of the drug.
You can also see that if everyone were identical, you wouldn't be able to have sympathy because a species without even the approximation of self-control would die immediately.
P.P.S. Sometimes empathy is split into affective empathy and cognitive empathy. This is cope, because affective empathy is nothing more than reflexive sympathy, and even female scholars realize sympathy is inferior to empathy. Being surrounded by women all the time, women are in a position to know that from experience.
Copying a bit of feminine sympathy can let men skip a couple cognitive steps. If you reactively feel what they feel, it saves having to work out
what they're feeling based on what they're thinking. Men often imagine this is what women also use it for. "Oh, she's sensitive so even when I'm not sure what he's thinking, she can figure it out." Haha! No.
P.P.P.S. That said, were everyone identical, you also wouldn't need sympathy because babby's first empathy would be automatic and accurate. They really would be you so whatever you're feeling about what they saw is also what they're feeling.