Children do notice it. Especially when it's their race that is being depicted negatively. Imagine how your grandchildren would feel if they were watching a Disney movie that said they were "cunning but not intelligent" or that they were "barely even human".
In my opinion this isn't really something that adults are pushing or going to great lengths to find offense. It is pretty blatant and kids do notice it and ask questions that parents need to be prepared to discuss.
Your examples were not sexism or ableism though. If they were actually doing those things, sure that would be just as wrong but you know that it's not the same thing. You're making a huge stretch. Hook being a bad guy is absolutely not the same as saying all disabled people are evil.
I'm genuinely curious if people are less bothered by this because there is this mythical Indian stereotype and they are typically not viewed as "real" people. I find it very hard to believe that anyone would be okay with their children getting up on stage and doing ridiculous accents, wearing stereotypical costumes, and singing racial slurs about African Americans or Asians. I think most people would be uncomfortable with that and would think it was wrong but when it's Indians its "just playing".
I also think shifting the idea to people being offended makes it easier to shift the responsibility to them ("they should just get over it and not be offended") rather than actually taking a moment to evaluate whether the thing is actually racist/offensive. Maybe instead of telling people that they are being too sensitive, we could just stop defending racist content as cute or benign.
Personally, I love Peter Pan and find that it evokes a sense of childhood nostalgia and wonder. Heck, my dog is named Tiger Lily. But I can simultaneously recognize that there are elements of it that are really racist and wrong (seriously, "The Great White Father"). There are many wonderful pieces of art and literature that are a product of their time that we can appreciate some aspects of them, but no longer find certain ideas appropriate. This has been normal throughout human history. Ideas and beliefs about what is acceptable behavior and language change over time. Our views on women and people of different races have changed drastically in the last 100+ years, so why is it so difficult to acknowledge that some things in a play written in 1904 may be outdated and racist when viewed through our current standards and beliefs?