I'd say you are generally correct, but the bolded part is the problem.
If you never show anything except the stereotypes, if you never present anything to people that is more nuanced, how are they supposed to know you are taking a shortcut?
There are still people who believe that all Vikings had horned helms, even though we know that is not true, but that was the only depiction they ever came across, so they had no way of knowing it wasn't accurate.
Therein lies the problem.
But at what point is something a generally recognized and incorrect new thing, and at what point is it a terrible stereotype?
Let's take the Vikings example. The single most influential example for the majority of Americans has been, and will continue to be, the Minnesota Vikings. Who have, wait for it ... helmets with horns on them. And mascots and fans with horned helmets. That's ... well, that's a thing. And it continues in all sorts of depictions.
On the other hand, most (not all, but most) modern depictions of Vikings, such as ... the show
Vikings does not have it. At this point, there is clearly a divide between the false trope of the horned helmet, and actuality, so much so that it's almost a cliche for someone to always say, "Hey, did you know that Vikings didn't have horns on their helmets? Let me explain that to you while I concurrently explain to you why I don't have a TV, why vaping isn't bad for you, and how bitcoin will solve all of the world's problems." Ahem.
Anyway, there is always room for nuance, but we also have to accept that some things aren't nuanced. Quicksand is neither as prevalent as some 80s media would have you believe, nor does it work that way. Legal cases do not work in the same way that they are portrayed in Suits. It is fairly difficult to knock someone out with one hit. Greek and roman statuary was not white back when it was created. Martial arts do not function in the way that they are portrayed in Western ... or Eastern ... media. The sanitized leprechaun we think of in America is a bit different than the traditional Irish one. And so on.
Life is always more complicated. And no matter how complicated and nuanced you want to make something, I can guarantee you that someone, somewhere, can make it even more complicated and nuanced.