When I first saw the movie I didn’t know who Beaver was, what was Jive, who Kareem Abdul-Jabbar was and lots of other cultural references.
I almost died laughing.
I guess you’re not a Baby Boomer, then? 
Not a Baby Boomer, not from the U.S., didn’t even speak English at the time (I saw it dubbed in Spanish), didn’t matter, the movie was that good.
The joke works with any white-haired little old white lady. But yes, given that you need a white-haired little old white lady, Beaver’s mom was the ideal choice.
Oh, granted the scene had additional depth to those who recognized the actress. My point is solely that that’s far fewer than “everyone”.
I’ve always been somebody who does not notice, or care, who the actors are. With very few exceptions (John Wayne maybe?) I have no sense any actor has a coherent body of lifetime work I should care about. IMO, if they’re doing their job, I, and you, should see only the character they’re portraying.
I got the impression during that movie that a lot of bit parts were played by big names. Which is a common enough Hollywoodism, and usually played for incongruity or as a joke. But which big names? Hellifinew.
As @Frodo demonstrates, the humor was so thick, so fast, and so silly-good that even if you missed half of it, the movie was still comedy gold.
I loved it.
The two actors not only wrote the script for that scene, they coached her on her pronunciation/inflection/accent(?).
Wow, in Spanish! Translating Airplane! so that the humor was retained must have been a daunting task.
So much of the movie (and many others) depends on cultural references. For instance, have you ever seen an episode of The Untouchables (1959–62), in which Robert Stack played lawman Eliot Ness? If you had, you’d know why I laughed so hard when he whipped off the sunglasses as a tough airline executive. And to me, Lloyd Bridges will always be “civilian frogman” Mike Nelson from Sea Hunt.
Such associations may not constitute all of the humor, but they certainly contribute to it.
And yes, I believe there’s a clip on YouTube where Barbara Billingsley talks about being coached in how to deliver the Jive dialogue. She says it was delightful and a lot of fun.
Yeah, the “I speak Jive” joke is largely lost on anyone younger than Gen X. Which is a pity, because that was probably the best joke in the movie.
I can just imagine the call “Hello, Central Casting? Send up a white-haired little old white lady who can be taught to speak Jive!” 
They did a great job, tough I must confess I don’t remember much, the last few times I saw Airplane! it was in the original language.
Many cultural references are shared anyway, from the top of my head: the man cleaning the front windows of the airplane, the cigarette the autopilot smokes after being… inflated, things like that are sort of universal, only the more intricate references were untraducible.
If they remade that scene today, I could see two older teens doing it.
“Bruh, whatever I just ate was so cheeks, I am literally cooked. My stomach is crashing out, fr fr.”
And then some white-haired old lady could come to the rescue. “Miss, I can help. I speak Zoomer.”
I dunno… You don’t need to be familiar with Jive, specifically, to recognize that the black characters are speaking in a thick, heavy slang typical of African-Americans. The joke stems, first, from treating Jive as a language, and second, in having the most incongruous character possible fluent in that language.
Actually, come to think of it… I moderate the D&D club at my school, and this past year, I was helping some new players (current high school students) make new characters. One of them was looking at the rogue class, and asked what “Thieves’ Cant” was. I told her “It’s basically slang so thick that if you weren’t raised on the street, you have no hope of understanding it.”. The light bulb went off over her head, and she said, “Oh, I speak Jive!”.
I get that. But I mean, specifically, the word “Jive.” I honestly don’t think it’s been used since, oh, the mid-80s? She could have said, for example, “I speak Ghetto” and a modern audience would have gotten it more readily.
The things that you mention (cleaning the windows, smoking after sex) are indeed universal because they’re physical comedy. This is why silent movies lasted so long. No dialogue was needed, and they could appeal to any audience, anywhere.
One I didn’t learn about until recently - the guy in the taxi was some big Calif anti-wasteful spending politician. His second scene (after the end credits) is hilarity, but it would have been that much funnier if I knew who he was.
* because of the credits / late scene in that movie I stay to watch the credits in other comedies.
But there could be cultures where the “smoking after sex” cliché does not exist for example, beyond the physical there is a shared,“western” for lack of a better name, culture.
[After he walks out unharmed after falling straight into a blazing fireplace]
“My skin’s made of asbestos!”
I think the best example of this is It’s a Mad, Mad, Mad, Mad World, which I saw at least three or four times back in the 1960s. Some critics have maligned it, saying that it was too long and overinflated. But I can say that part of the fun was waiting to see just who would turn up next on screen. (For example, Jack Benny, Dick Shawn, Doodles Weaver, The Three Stooges, Eddie “Rochester” Anderson, and Peter Falk.)
Anyone younger than me at the time (I was eight when I first saw it) would probably not have recognized some of the featured players if they weren’t as immersed in popular culture as I was even then, and I’m certain virtually all of them would go over the heads of the generations that have come since. This is a shame, since they were all masters of comedy, physical and otherwise, and deserve to be remembered.
Maybe they just don’t bother to look?
I can’t find a cite, but I believe they originally wanted Florence Henderson, who played Carol Brady.
Additionally, Pete Rose was originally going to be the co-pilot played by Kareem Abdul Jabbar.
Jabbar did it in exchange for enough cash to buy an expensive rug.