Yep. I'd add that that the idea of a minimum number of observations isn't about making sure the CLT will "work". Take a case where you are calculating the mean of a set of independent random variables. What the CLT says is that as the number of variables in the set of independent random variables you're averaging increases, the sampling distribution of the mean will converge towards a normal distribution. When people say you can invoke with CLT with X number of cases, what they mean is that with this number of cases you can be reasonably sure the sampling distribution of the statistic will be approximately normal - due to the CLT. It's not a case of the CLT being invalid with small sample sizes and valid with large ones.

Definitely good to mention. That's mainly why I used "works" to imply a fast and loose, but likely more common and less correct, interpretation of it. I think people tend to miss the idea that it's not black and white, but rather has to do with the appropriate use of a normal distribution for inferences. In other words, is the sampling distribution you're working with for that fixed sample size

*reasonably* approximated with a normal distribution? If so, we can use some more familiar approaches. If not, we lose some nicer properties and need to look elsewhere for some support. It seems to me like a lot of this gets lost on many people and they boil it down to black-and-white thinking as they've done with the rest of their stats knowledge (if some of it is lost on me, I'm currently unaware, so feel free to point it out!).