It doesn't exactly work like that, though, does it? Negative stereotypes don't necessarily "directly" affect people in the sense I think you mean. They have a more gradual, long-term, compounded effect. And it's often not an entirely obvious effect to observe at the level of a single person, because often the effect is to influence a person's perception of what roles (in terms of work, home life or anything else) are reasonable choices for them, and it's not easy to perceive when someone just doesn't even consider doing something because they've learned over time that it's not a thing that People Like Them do. After all, *most* people don't become astronauts or firefighters or Olympic athletes, so it's hard to look at *one* person who didn't do that and say "hmm, maybe media stereotyping played a role in this". You have to have a more sophisticated analysis.
One case I've found really interesting lately, which maybe isn't one you'd expect, is the show American Ninja Warrior in the U.S. It's an extremely popular sports-reality show (involving extremely fit people doing extremely hard obstacle courses), and to its credit it's made a conscious effort to promote female competitors. It's really fascinating to see the number of kids who see a woman doing well on a show like that and are inspired to take up the activity for themselves. I've seen more than one girl say something along the lines of they just didn't know it was *okay* for girls to be strong, muscular and powerful before seeing ANW or something like it: they just didn't see it as a choice. And indeed if you think about it, someone like Meaghan Martin (look her up, she's amazing) isn't a common sight in the media; if you think about the stereotype even of a 'fit' woman, it doesn't look like her. There's an overlap with tennis and all the shade that gets subtly thrown at players like Serena Williams who are unapologetically muscular and powerful; there's a strong current of belief that even elite female athletes must be somehow 'feminine', i.e. slender and pretty.
To put it simply: of *course* what you see around you, in the real world and in the media, affects your idea of what you yourself are capable of and 'allowed' to do, especially at the young ages where people often form their goals. It would be surprising if it were otherwise, wouldn't it? There are also of course obvious potential downsides to allowing what is effectively censorship, but I think it's nuts to deny the idea that widespread stereotyping can have this kind of effect.