Robert A. Heinlein famously said in "Time Enough for Love"
A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly. Specialization is for insects.Virtue is like that; though many are more common in one gender than another, none are the exclusive province of one gender, and in my view as a Christian the lack of any virtue will not be excused by the idea that's the other gender's speciality.
So, for example, I think Lieutenant General Sir Robert Baden-Powell, for example, was more a real man than almost anyone ever, but at the same time more of a real woman than, say, Twisty Faster. And Elizabeth Moon, judging by her characters like Paksenarrion or Heris Serrano, is far more a real man than Dudequest as well as likewise more a real woman than Twisty.
Given that these virtues are not exclusive to one gender or the other, why identify them with gender? That can be embarrassing for a girl that has better-developed "real man" virtues or a boy that has better-developed "real woman" virtues. To go a little further, it can lead to people who, lacking any virtues at all, try to pretend their very lack of the opposite gender's traditional virtues is in itself a virtue. E.g., a man with no compassion who claims he's not a pussy or a woman with no courage who claims not to be poisoned with testosterone; that man probably has no courage and that woman probably has no compassion.
The reason not to get rid of the gender identification is that we just don't have a good alternative way of inspiring or teaching these virtues yet. And it's much better for society if the virtues are divided half-and-half than if no one has them. The perfect is the enemy of the good. If the branch you are sitting on needs pruning, move to another one BEFORE you saw it off. It's ridiculously easy to see the flaws in the old system, and damnably hard to create a new one that's actually better. Experiments with significantly different moral teachings on the large scale tend to result in more death and misery than anything else in history except maybe the Black Plague. Meanwhile, recognize and praise courage in women and compassion in men as well as the other way around, there's lots out there to see and that'll get you farther towards finding a better, less gendered way to inspire courage and compassion.