I am also very supportive of women's rights but lying is not helpful.
Worth knowing: although they attract a lot of anti-feminist losers, the "men's rights" activists are absolutely correct that men do not universally have the same support programs or even legal presumptions that women do. These can vary widely from state to state and even from court to court.
It's not nearly as big an issue as "they want her to die from a miscarriage", but "they presume he's the inferior parent" or "they presume he caused the violence even if he's the one bleeding" are also sexist oppression.
(Comparisons to the anti-woke "all lives matter" bullshit are apt -- men can and should recognize that relatively minor slights and injustices are not nearly as urgent as denying pregnant humans life-saving care!)