I have talked to entirely too many women who have had men cry during breakups. The crying is immediately followed by emotional manipulation and, in many cases, verbal abuse.
I think one of the reasons men are socialized to not cry is because this behavior just isn’t a good look on them.
I have cried a handful of times in my life and I have always sought out somewhere private to do so. I have never seen anybody, male or female, cry in public or in front of somebody where the witnesses wound up with more respect and not less for the cryer afterwards.