It's not a myth: The programs were expressly designed to push Black men out of the home while keeping Black women beholden to the Democratic party because they hooked them up with food, a place to live, and whatever other services they got.
I told y'all before, the summer of '97 I was crashing on my best friend's couch (the woman I would end up marrying 4 years later) but the rule was I had to keep my stuff in the basement and it had to look like it was just being stored there 'cause she was on section 8 and getting WIC for her 3 year old 'cause if they did an inspection and determined I lived there she would get kicked off. Sure as shit, about 3 weeks after I moved in, she had an inspection and when they questioned her about my stuff in the basement, she told the worker that it was just being stored there and they let it go.
To your point: Yeah, that's how it was at least up through the early 00's. I don't know how it is today with those programs, but in the past it was walike that.