I would argue the opposite. Each field of study has so much more depth now that on any given topic there is more that a layperson has to learn to be on par.
Lay people do tend to be much broader generalists (at least in terms of scientific understanding) than they were in 1939 but they are further behind the specialists.
But broadly the gap between scholars and "the public" is as massive at it's ever been. Meet a few members of "the public" socially, outside of tech or academic circles or however you usually meet people. Go to some random bar or club or whatever and make some friends. I think you'll find, like I do, that most people basically don't know anything. They can sort of do 3rd grade arithmetic, but struggle to apply it, and don't know anything else about mathematics - most people don't understand how to use negative numbers or what they're for, for instance. They don't know anything about history - they don't really know what the Renaissance was, or when the Roman Empire was, or when and where human civilization began, or anything else. They absolutely know nothing about physics - they are completely unaware of the idea that acceleration due to gravity is not dependent on mass, which I see as the starting point of teaching physics.
Swap the places though and send someone who's held various mechanic, repair, manufacturing roles to a high end cocktail bar full of academics and let them ask questions on topics they know, and they'll be shocked (or maybe not) at how most of these academics basically don't know anything about how the nuts and bolts machinery that keeps the world running actually functions in real life, and certainly couldn't do most of the work themselves. They'd probably also come away thinking they'd rather have a random person from their dive bar in Nebraska help them for a day at work on a hard project than one of the people in the fancy bar.
Without being able to rely on these masses of the un-academic "public" to run the world they live in, the academics would largely be helpless and their knowledge useless as they starved to death as the world crumbled around them.
Specialized knowledge and training is amazing, and it allows us to advance as a civilization, but I don't think the average "academic" is much more intelligent or capable in the raw sense than the average person in "the public". Both are needed for society to function and progress. Without the academics there will be no progress, and without the public it will all break down.
Of course, there are also people who have little useful knowledge / skills / motivation either practical or academic. These people can however just as easily exist in the social / family circles of the elite, as in those of the public, so I don't think it's fair to throw them in either group for judgment.
But once a person's intelligence gets too low, they can't figure out anything. But, they can still learn. They don't see anything wrong with them, they think it's how the world works, people know and can do things that somebody taught them. When they see somebody figure something out, they think it's something they learned in the past, but are dumb and evil people who laugh at them for not knowing. Stupid and evil peoppe who refuse to pass on what they were taught. And they never figure out that they aren't normal, or even think that they're brilliant, because they remember more than others.
Today, the gap between scholars and the public is probably much narrower. This is certainly the case in physics, and I would think in biology and chemistry as well. I believe the only fields where the gap has significantly grown are computer science and pure mathematics, simply due to rapid progress in those fields.