It's really interesting that such a bland, un-subversive show whose only mentions of sensitive topics are in bad throwaway jokes is so heavily censored. I guess a more interesting show would just not get aired at all.
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For example, the joke about the Chinese restaurant ("I'd be more concerned about what they're passing off as chicken") plays off of the stereotype that Chinese people eat dogs and cats, and the “passing off” remark implies that the Chinese restaurant owners are deceptive and would immorally and illegally serve their guests a different kind of meat than advertised. I can definitely see how that joke would be considered offensive.
The author labels that joke as "harmless" but you don't have to be a Chinese censor to interpret it as reinforcing harmful stereotypes. I dare you to show that scene at a liberal college and notice how few laughs you get.
Similarly, the racist remarks about Chinese people made by Sheldon's mom are somewhat offensive if taken at face value. I guess the joke is supposed to be at her expense instead ("old people are racists" is an American comedy cliche, if a somewhat tired one) but it's conceivable that either the censors didn't get that, or they feared that their audience didn't get that, so they decided to cut it out entirely.
"They wouldn't get that" is probably also the right explanation for censoring the joke about Jews eating at Chinese restaurants during Christmas, which is a very American tradition. That doesn't imply the joke needs to go, but I can see how that would, at best, leave Chinese viewers scratching their heads.
Did you see the recent video where the white guy dressed up in a poncho, big hat, and fake mustache and carried around maracas? He asked a bunch of white kids on a college campus if they thought his outfit was offensive to Mexicans, and they all said yes.
Then he went to the Mexican part of town and asked actual Mexicans, and they all said it was funny or that they liked that he was trying to honor their culture. Not one of them was offended.
So perhaps it would be good to ask a Chinese person if this joke offends them.
And not all racism / bias is equal. Maybe you are right that Chinese and Chinese-American people would not be offended by this, but it seems completely reasonable that they would be, and the onus on you would be to get data that they wouldn't. It really doesn't matter what liberal college students think at all, unless they happen to also be of Chinese or of Chinese descent (or they are southeast Asian, and tired of lazy racism that doesn't bother to distinguish such things).
edit: it was in fact PragerU (https://en.wikipedia.org/wiki/PragerU) which is intended for entertainment. It should not be considered reliable or unedited.
FWIW I have a few data points -- this is something my Chinese wife has literally said inside a Chinese restaurant, and some of her other family members have said similar things about not trusting that the food being served is what they said it was.
I don't know the right answer, but I definitely think it would be understandable if someone didn't appreciate that joke. And worst of all, it's just in service of the cheapest, blandest kind of humor. The writers should be ashamed of such lazy work, regardless of bigger issues. "Would it work without a laugh track" clearly fails badly here, as it does pretty frequently in TBBT.
Chinese folks being weary of restaurants with swapping ingredients for lower tier is not comparable to assuming chicken being swapped for cat, which is a tired joke. Usually reserved for pricer seafood, hence pick your victim tanks. Many restaurants do similar type of substitute shenangians, like I'm pretty sure the hipster burger joing is not serving genuine kobe beef patty for $15, but they're also not serving ground chihuahua either. Like even in PRC you're worried about things like gutter oil at a hole in a wall joint versus slightly cheaper grade of sea cucumber at a fancy restaurant. Even during the pork crisis, no one was particularly concerned that restaurants were feeding them cat/dogs instead.
E: relate back to your parent comment, there's somethigns like cultural appropriation that most (especially older gen) Chinese don't care about, i.e. they thumbs up for white girls wearing qipao.
Isn't PragerU a far right site know for promoting bizarre things? I'd would definitely call it "unreliable".
https://www.usatoday.com/story/news/nation/2013/02/20/fish-s...
https://projects.tampabay.com/projects/2016/food/farm-to-fab...