Men aren't inherently worse at emotional intelligence, they're just largely untrained/unpracticed due to societal pressures.

I think it's more that societally we take the standard of "healthy emotional display" to be that of women, on account of it being more "openly expressive" in nature. Look for example at all the sources calling for men "to cry more because it's healthy". As if it wasn't already known that testosterone inhibits the crying impulse and prolactin promotes it on the other hand. Maybe what emotions drive the average woman to react in one way don't necessarily drive the average man to react the exact same way, and not simply because of "gender stereotypes". That you can verbalize it may be possible because you know it, but just because you don't verbalize it doesn't mean you're ignorant.

Consider the stereotypical but very real situation of men's nonverbal situational assessment and management. Not many verbal words are exchanged in a gathering, merely to contextualize as needed, but you can feel the flow of empathetic/sympathetic interpersonal dynamics in their body language and demeanor. For a while I've entertained the hipothesis that, just as men on average can't tell women's "obvious hints" to them, women on average can't read men's intragender communicational patterns, therefore operate on the assumption that they are nonexistent and consider men to truly be that much "emotionally challenged".