Having worked in childcare / early childhood education, I've seen it a lot. Kids younger than about 5 or 6 don't care. The boys love to play dressup and the girls love the sandbox with the toy cars. As they get older some start to fall into stereotypical gender roles... and even then it's usually obvious that they're learning from / being indoctrinated by their parents. My use of the word indoctrinated is only to indicate that sometimes these gender roles are passively learned from the parents example, but in some cases there are parents that actively enforce / teach their kids to follow gender roles. For example, I've seen a father, on arriving to collect his son, react almost with horror to see the boy in a skirt playing in the mock kitchen.

Cliche is the wrong word. Gender roles in society are a topic that spans many disciplines and has a whole lot of detail to discuss. They are a huge thing, and you're right: the gender role assigned to girls does prescribe for them a lot of child rearing practice. That said, after seeing what some kids will do to a baby doll, I can confidently assert that such play is no substitute for actual instruction.