I absolutely agree that this will help people stop being confused about Godel's theorem, I just don't know why EY does it in this particular post.

Do you have any basis for this claim?

Nope, it's pure polemic ;) Intuitively I feel like it's a realism/instrumentalism issue: claiming that the only things which are true are provable feels like collapsing the true and the knowable. In this case the decision is about which tool to use, but using a tool like first-order logic that has these weird properties seems suspicious.