I picked the first three results from your search and they don’t support your argument at all. They even call it out as a stereotype (which it is).
Male leaders received lower effectiveness ratings when expressing sadness compared to neutrality, while female leaders received lower ratings when expressing either sadness or anger.
So sadness for leaders is neither ok for women or men.
The analysis of emoticon (emotional icon) use in online newsgroups appears to reinforce the stereotype of the emotional female and the inexpressive male until further examination suggests otherwise. The most interesting finding of this study is illustrated by the pattern of change that develops for both genders when they move from a predominantly same gender newsgroup to a mixed-gender newsgroup. The changes that take place in emoticon use when moving from same-gender to mixed-gender newsgroups indicate that rather than the emotional expression of females being silenced or muted by male encoding of emoticons, males adopt the female standard of expressing more emotion. Furthermore, women have added dimensions including solidarity, support, assertion of positive feelings, and thanks, which were absent from the male-created definition of emoticons and their use.