Well spoken, Sunny, but you left something out. This was an intersectionality between racism, sexism, and transphobia.
These bigots don't just think Michelle is a man; they think she is a trans woman, which to these racist, misogynistic, transphobic bigots, is the same thing. Consciously or not, they view trans women as traitors, men who have rejected masculinity, and that feels like a personal insult to them.
Calling Michelle a man is just one facet of transvestigation, which GLAAD calls a "hate-driven conspiracy theory [that] maliciously targets cisgender public figures from Frances First Lady Brigitte Macron to Michelle Obama to Olympic boxer Imane Khelif and then investigates them offering fake pseudo-scientific evidence that they are transgender (with the underlying bigoted and ignorant implication that being a transgender person is a bad thing). This is a notable example of how transphobic hate impacts everyone as the targets of the harassment are cisgender people. The trope has even been weaponized against right wing figures such as Kyle Rittenhouse and Andrew Tate. Disinformation researcher Dr. Jay Daniel Thompson describes transvestigation as a mash-up of QAnon, celebrity gossip and the myriad anti-trans conspiracies. Source: https://glaad.org/transvestigation-definition-meaning-anti-lgbt-online-hate/
These people comb the internet for photos of famous women they don't like, looking for "clues" like broad shoulders, a prominent forehead, or a strong jawline that they regard as "proof" that the women in question are really trans women (who they call "men" because they either don't understand the distinction or have chosen not to recognize it).
If a woman they are targeting is married to a male public figure, these accusations are used as slurs against the male partner (becaise if his partner is a "man,' he must be gay.
)
Never mind that there's absolutely nothing wrong with being either gay OR trans; these people believe that being either of those things is wrong in every way, and they delight in spreading their lies. Transvestigation is a sick, disgusting activity, and those who pursue it are sick, disgusting people.