Leftist have this great argument, I did not chose to be your child so I owe you nothing and you must take care of me.

I've seen this and tried to seriously engage it. Not choosing to be born is pretty much a non-point unless you want to make some esoteric buddhist argument.

"I owe you nothing" is a little more involved. It necessitates some form of contempt or abandonment. If your parents didn't raise you at all (like they were in jail and you grew up in an orphanage), then I could accept the claim because it'd prove that the parents did not fulfill their roles. If your parents at least tried, however, you cannot deny them being your parents because to do so is to deny the influence they had on your development, ultimately forcing one to deny themselves. This part actually clicks for leftists to me, as they provide no evidence that they either accept themselves or are happy with themselves.

"You must take care of me" doesn't quite follow unless you take a jump from the previous point to them actively hating themselves and hating their parents for giving them a life of misery and contempt. It's clear the context is of an abusive sort of manipulation.

If the whole phrase were "I did not choose to be your child and you must take care of me" I could accept it as a benign appeal to nature, as it'd be bizarre for a parent to not want to take care of their child - though I would assume that there are some hard limits on being taken care of, such as no abuse tolerated and support not being infinite.

You're right that it's a good representative saying.