A century isn’t that long a period of time, and there are actually tons of small but noticeable changes in dialect between today and those old radio dramas that you probably just aren’t hearing.
If you want to focus on spoken language, as per the claims about sound recording, Simon Roper and Dr. Geoff Lindsay both have videos on YouTube about dialectical changes over time. My impression is that the difference between 1924 and 2024 seems about as big as the difference between 1824 and 1924, despite the invention of sound recording. And this is exactly what I would expect. I don’t think you could name ten pre-1940 films that most Americans have seen, for instance.
https://www.britannica.com/video/187707/David-Crystal-pronun... for how it would (probably) actually have sounded like. Pretty hard to understand!
Conversely, I don't think I know any Americans who haven't seen The Wizard of Oz (1939). And again with the sole exceptions of superficial idiom and vocabulary (and some really minor pronunciation changes that are as much theater affectation as they are language) there's absolutely no change at all from the stuff you hear from a 15 year old member of the High School class of 2027 (I live with one). Again, I really don't think that's normal historically.
And I mean...actually listen closely to these people's accents. You're really going to claim that there's "absolutely no change at all" between them and your teenager? https://www.youtube.com/watch?v=amclN9RG49c
I'm sure a lot of modern English speakers have a much better ability to comprehend dialogue in English Circa 1700 due to familiarity with Shakespeare than they would otherwise, but influencing modern usage is a very different thing.