Women say they want a man who's emotionally intelligent.

I don't think this is true. What women really want is emotional validation. They want men to display emotions that align with and validate the emotions they are feeling. Men's emotions are viewed instrumentally rather than being accepted as part of the men holding them. "Emotional intelligence" is merely a way of laundering that desire into something that is less obviously selfish. The reason so many men seem emotionally stunted is that they have been socialized to accept such instrumentation of their emotions by others and "stoic" suppression is a way of bearing that emotional burden. Of course, such "stoic" suppression doesn't provide the validation that women are looking for, leading to the use of shame to drive men away from such relief.