I don't think such a period ever existed. Hollywood had the Hays code, comics had the comic code etc. Popular culture always have to navigate the hangups and sensitivities of society.
It is possible that these concerns were invisible to you because they corresponded to your own values. When they change because society changes, you suddenly notice them.
However, by cultural sensitivity I refer to something beyond codes. Nothing that this article points out is particularly illegal, immoral or even shocking. Yet, probably due to cultural globalisation, social media, and the polarisation of the debate in the US on certain social issue, I am under the impression that this kind of clumsiness is more likely to be noticed.
Probably not in the US, no. Then again, the world is comprised of many other locations and cultures. Even in the West, I'd say mainly predominantly protestant countries (among which the anglosphere as a whole) tend towards political correctness.
It sounds like you're using it more like a Zeitgeist?
You can call it "group-think" or you can call it "cultural values".
I was responding to the suggestion that there was a time and place without "political correctness". I dispute that. Every time and place have some ideas of what is and isn't morally acceptable. But if you are born into it, it may be invisible. The majority of the audience at the time did not wonder why there were so few black cowboys in movies. But they noticed it when more black cowboys started to appear.