Confusion in the added loop of the Halting Problem
What we show is that this augmented $H$ can not even exist. You gave me $H$, and I gave you back an augmented $H$, called $\hat H$. We can build this $\hat H$ by adding the extra loop and if statement, but its easy to show that this augmented $\hat H$ will never be able to exist: $\hat H(\langle \hat H\rangle)$ is not either rejecting or accepting, nor stuck in an infinite loop, but we know that any TM must be either one of those.
Therefore, we get a contradiction to the existence of $H$: we showed that if $H$ exists, then some other $\hat H$ exists, but also $\hat H$ couldn't ever exist.