But then why do these stereotypes remain stable across generations?

Not defending the progressives in general here, but there are two very simple explanations for your question.

1) Some stereotypes don't remain stable across generations.

For example, I heard that in the past, pink was considered a "boy color" and blue was considered a "girl color"; or that it was believed that black people would be bad at sport. So, some stereotypes change and some don't; and we would need a meta review to find out whether there is something special about those unchanging stereotypes, or whether it just means that if you flip a coin two or three times, sometimes you will get the same result repeatedly.

2) If a stereotype already exists, it is more easy to keep believing in the existing one (confirmation bias) than to invent a new one.

(Disclaimer: None of this is meant as a general proof that all stereotypes are incorrect. It's only an explanation of how a stereotype that happens to be wrong could remain stable across generations.)