> mostly because we don't see similar things pushed on overweight men ('manboobs')
Because moobs are just balls of fat. Actual breasts contain glandular tissue which moves differently and has more heft.
While men with ordinary moobs don't need bras, men with gynecomastia (actual breast tissue growth) really should be wearing them. I'm more interested in destigmatizing the notion of gynecomastic men wearing bras than I am in discouraging women from having support.
Also, here's something I can actually give insight into specifically because I'm transgender. Before I transitioned, I was obese (and I still am), and I had noticeable moobs. I never felt like I needed a bra back then, as there was no pain or discomfort from moving around. After being on HRT for a while, I developed significant breasts, and now it's extremely uncomfortable and a fair amount painful for me to go without support. Moobs are not boobs.