Curiosity:
Allow me to modify your scenario slightly in order to isolate the issue under discussion. We have a guy who learns before Shacharis and after Maariv. The financial situation of the family is such that between Shacharis and Maariv he will not be learning. There are also things that need to be taken care of at home (e.g. taking care of the kids, cooking, cleaning etc.) which will take up the time between Shacharis and Maariv. Now there are two parents in this family and there are two jobs which need to get done during the same period of time (i.e. only one person can do each job) – the job in the workplace and the job in the house. Now let’s say that the husband is better at cooking and cleaning and taking care of children, and the wife is better at making money. Well according to the Fifth Principle of Economics, the productive way to run this household would be for the woman to go out and work and the man take care of the home. Is there anything wrong with this picture?
Furthermore, much of what is conceived of as men’s roles or women’s roles, is based on men in general being more _____ and women in general being more _____. Now everyone knows that in general men are taller than women and have more muscle mass then women. So obviously, if a married couple needed to get something done which involved height or strength, the husband should be the one to do it, right? But what if the wife is taller or stronger? Should it still be the husband’s job to do it simply because in general, men are taller or stronger than women?
Similarly, it may well be that women are generally better at cooking, cleaning, and childrearing, and are more emotional and empathetic and many other things. But that doesn’t mean that any given man can’t be better at those things and have more of those traits than any given woman. So why shouldn’t the husband do the cooking if he’s better at it and enjoys it more?
All this was just to show that there is no such thing as an inherent role. However, there are some exceptions. Some roles are Divinely dictated, e.g. a man has to do Time-Bound Positive Commandments, and some roles are physiologically dictated, e.g. women giving birth (at least for now). But on anything else, why should we create artificial limitations?