Recall that Grothendieck's Homotopy Hypothesis states that the homotopy category of weak globular $n$-groupoids (as in https://arxiv.org/pdf/1009.2331.pdf) is equivalent to the category of homotopy $n$-types.

It's also well known that strict globular groupoids are not enough since $\Pi_3 (S^2)$ cannot be rectified to a strict $3$-groupoid due to the non-vanishing of the Whitehead bracket.

On the other side, it's well known that every quasi-category can be rectified into a strict $(\infty, 1)$-category. More generally, there's Berger-Moerdjik result that says that a bunch of algebras over operads can be rectified. In particular, $A_{\infty}$-algebras can be rectified and also homotopy coherent diagrams (Vogt's theorem).

In view of these observations, I have the following questions:

1) By considering $\Pi_3 (S^2)$ as a quasi-category (i.e., extend it by degenerated simplices), we can rectify it. Why doesn't it contradict my first observation? (**EDIT :** As Yonatan mentioned in the answer below, only the level $0$ and $1$ can be rectified. Therefore there's no contradiction.)

2) Why can't we rectify weak globular groupoids? By Vogt's theorem, homotopy coherent diagrams can be rectified. So, what fails if one views a weak globular groupoid as a homotopy coherent diagram and, then, rectify it?

Further comments about the intuition of why one cannot rectify weak globular groupoids would be also of great utility.