The sexist remark about Barbie is the set up for the self deprecating joke. It's not said seriously, it's there to make you feel a little bit offended and engaged to then realise that they are making fun of themselves.
I've heard almost this exact thing said unironically to no laughs where people believe they're too smart to have girlfriend so it just strikes me as either a tone deaf joke or a cringe real thought so it doesn't land.