It's not obvious that trying to catch every mistake yourself would be the optimal division of labor, assuming that you really are in an environment where you can trust on others to correct some of the mistakes.

I agree that we could probably rely more on others to catch our mistakes in certain contexts where equal expertise can be assumed. The problem is, if you're writing an article or a book for example, you're usually the expert compared to your readership, so you can't really expect others to reliably correct your mistakes, and some of your mistakes get cluelessly adopted.