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.
So... you support government censorship of jokes that somebody, somewhere might be offended by?
Edit: it seems it's actually relatively easy to find jokes that are genuinely offensive and degrading in PG rated films. Why that's considered less potentially harmful to kids than showing sex between consenting adults I honestly don't know.
Most channels not restricted by those rules (subscription cable & satellite) set in-house standards on content for commercial reasons. And of the broadcasters that are covered by the regulation, they are the old stodgy networks and never choose to get near the boundaries.
The FCC can and does regulate over‐the‐air broadcasts to a stricter standard, thanks to its exclusive authority over the inherently limited wireless spectrum. It restricts not just obscenity, but indecency (explicit sex) and profanity (bad language). However, this power does not extend to (e.g.) cable TV, which is not broadcast over the publicly owned airwaves.
The US really does generally have stronger free speech protection than the rest of the developed world. There is no equivalent in the US to a work being “refused classification” as seen in Commonwealth countries. The First Amendment would prohibit it. Some retailers won’t sell unrated or X‐rated films or AO‐rated games, but others can, because the ratings systems are formed by industry groups and are not compulsory.
When the Christchurch shooting happened, the New Zealand government banned both the shooter’s manifesto and the livestreamed video, making them illegal to possess or distribute. I doubt such a thing could happen in the US. (I remember my surprise that NZ actually has a government office named “Chief Censor.”)