This isn't some manosphere conspiracy. She's just a scorned lover. Likely, she knows people will find out eventually anyway. My bet is that she wanted to announce it to pique curiosity, knowing people will investigate, and she's banking on whatever they find being more damning for him than it is for her. Then instead of it being her going full bitch mode she's merely responding to people or clearing the rumors up. He wanted to keep it quiet, she didn't and this is her way of popping the balloon.
As for the stuff like "my" kids, well, if she's a stay at home mom and from her perspective taking on most of the responsibility for the children aside from financial then yes it will feel more that way to her and regardless of how based any of you think Dank is, likely that will only get worse now they're separated. As entertaining a personality he might be, personalities like that don't often tend to make stable and engaged parents.
Good luck and godspeed to the children involved. They'll need it.