I guess it depends on which end of the joke you are. And how often do you have to deal with it. It could be just a joke, or straight up bullying mascarading as joking. I am sure you know what I mean.
Others have already written about this extensively but there is a difference between humor that punches down and humor that punches up. There's also a big difference between the joke being bigotry and the joke being that a character is bigoted.
In the Airplane! cockpit scene the joke isn't that the pilot makes the stewardess and kid uncomfortable by sounding like a pedophile, the joke is that he's supposed to be a heroic character but acts like a total creep. It's a subversion of expectations based on the tropes of 1950s/1960s Hollywood movies.
Likewise the scene with all passengers lining up to slap the woman while shouting at her to calm down doesn't make fun of women for being hysterical. It makes fun of the sexist trope of women being hysterical and needing to be "calmed down" with physical violence. The sexism isn't what's funny, the sexism is what's being made fun of.
The problem with modern comedy is that a lot of political punditry has moved to taking the form of comedy and "it's just a joke" (or for a few years on YouTube "it's just a prank") has become a way to defend actual bigoted statements.
When a conservative pundit who is opposed to gay rights calls a gay person a homophobic slur or follows a mention of them with a caricature of them wanting to have oral sex with a lot of men, the "joke" only works if you share the idea that gay people are sexual deviants and bad. It isn't really a joke, it's just mockery.
The reason a lot of older comedians find it hard to adjust is not that comedy has changed. The mechanics of humor have largely remained the same, it's just cultural attitudes and politics that have changed. If your politics have remained the same, you'll find it harder to do comedy expressing those politics now than when they were more closely aligned to mainstream.
In other words, if you used to have a close circle of friends with misogynist opinions a misogynist joke may have gotten a few laughs out of them. If they've all grown out of it and matured in their understanding that women are actual persons rather than just objects of sexual attraction, your old jokes will no longer work on them. They may not actually get offended, but they won't be amused.
Jokes work because they carry a message through a combination of context, content and subtext. If that message is expressing support for an oppressive social dynamic, it can be bullying. It's that simple.
I agree with you. I like your statement "The sexism isn't what's funny, the sexism is what's being made fun of.". Similarly, in this article there's a line "a joke can illuminate uncomfortable subjects by giving us permission to laugh at them". At my corporate job, we have a focus on diversity and in these times of Black Lives Matter and COVID related anti-Asian sentiment, we've been challenged to think about how we talk to our kids about jokes. Here's my take...
I've tried to express this to my teenager - that sometimes these jokes that seem racist/sexist/ist are actually making fun of the absurdity* of racism/sexism/ism, but it can also be hard to tell sometimes, and many jokes truly are bigoted. As someone who hears an off-color joke, try to cross examine it before knee-jerking taking offense (although, often offense is warranted), and as someone telling an off-color joke, be very careful* in your delivery.
For jokes, there's a time and a place, and the flip side to that is that there's a wrong time/place too! For example, I like the joke "I want to go peacefully in my sleep, like my grandfather.... not screaming like the passengers in his car" - but I would never tell that to someone grieving the recent loss of a grandfather or someone who passed in a car accident. And given the current BLM movement and the anti-Asian sentiment - this is definitely not the time for some jokes. My teenager is mixed race, half Asian (prefer not to say which), and they and their friends sometimes feel like they have the green light to make certain Asian jokes. I've told them these are not the times to make those; in other times, it might be okay, but keep a lid on that stuff until some of this passes, because this is serious - it's not the time.
And very importantly: just because you find an off color joke funny as opposed to taking offense, it doesn't mean that you're a racist/sexist/etc... and it doesn't mean that you have a character flaw. But just because you find the humor in it, doesn't mean your insensitive or bigoted. As a society, we shouldn't be afraid to laugh and we shouldn't virtue signal by jumping to offense. But if someone takes offense to a joke that you make, you're almost always the one in the wrong (not always, but almost always, so do some self reflection).
Absolutely, the content of a joke matters as much as the context. This not only includes the setting and audience but also the speaker and the audience's understanding of the speaker.
This is especially relevant for in-group jokes and self-deprecation. A Black person performing for a Black audience can poke fun at Black culture as a form of self-deprecation, but if a white person tries to perform the same joke to the same audience the intention will get muddled, and if the audience is white it quickly turns from laughing at yourself to laughing at a marginalized group.
This effect is also why there is no such thing as the "n-word pass": even if a Black person tells you as a white person that they're okay with you using that word and even if they're honest, that only makes it okay in the narrow context of conversations between you and them. Even having another person around can quickly get messy.
Likewise "Karen" originating as a pejorative against white women "playing the victim" sounds very different coming from a Black person or a white man.
Language and communication are complex and jokes are just another way to communicate ideas with language, even if the ideas may be somewhat non-obvious.
Then the point remains in discriminating joke and joke. Especially in the current confusion and climate of manipulation, it is important that people are led to understand what is e.g. mockery of bigotry, vs "bigotry and mockery". Oversimplifying the world and its expression serves no one (it is damaging).
The only people who care about political correctness are liberals and conservatives, and the latter group more so than the former (which is why they are more likely to engage in "dogwhistling"). Political correctness is about the "how", not the "what".
Making jokes about Jeff Bezos is punching up. Making jokes about the homeless is punching down. The reason so many people got mad at Dave Chapelle is not just that he mocked trans people (which is a marginalized identity) and is cis (which isn't). It's that he thought being Black (which is also a marginalized identity) excused it because he completely ignored that Black trans people even exist. He assumed that as a Black comedian he was always going to be punching down regardless of whom he ridiculed.
The term "political correctness" among conservatives is often an expression of the assumption that everyone else is hypocritical and bigotry is the norm. But it's not about hiding your bigotry, it's about actually holding a different opinion. Political correctness is good in so much as it makes it harder to promote bigoted views, but it's insufficient when it comes to solving latent bigotry in a society or subculture.
I remember a recent discussion (on the slatestarcodex successor, I think) that a lot of 'punching up' is actually complaining about how (some) lower class people have too much money..