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.
Best Answer
First, let me remark that your question does not seem to be about the homotopy hypothesis, but about rectification. More specifically, the homotopy hypothesis concerns the question of whether (some version of) weak globular groupoids is equivalent to, say, Kan complexes, while the rectification problem you mention asks whether weak globular groupoids are the same as strict globular groupoids. The confusion then seems to arise from the fact that on the side of Kan complexes, there are some aspects which can be rectified. For example, we can rectify a Kan complex into a strict simplicial groupoid. One can then maybe rephrase the question as: given that we believe the homotopy hypothesis, how can it be that Kan complexes can be rectified while weak globular groupoids cannot? The answer to this question is simply that Kan complexes cannot be rectified either. Indeed, if one replaces a Kan complex by a simplicial groupoid, the mapping spaces of this simplicial groupoid will still be Kan complexes. This can be informally described as saying that we have rectified the levels of objects and morphisms, but we have left unrecitifed the level of homotopies between morphisms, homotopies between homotopies etc. all the way up. Note that by only rectifying the 0 and 1 dimensional levels you can still avoid the counterexample $S^2$. On the other hand, that's the maximum you can do. For example, $S^2$ cannot be modeled by a strict 2-category enriched in simplicial sets: indeed, the group of self equivalences of an identify morphism in any simplicially enriched strict 2-category is a simplicial abelian group, but the corresponding group of automorphism for $S^2$ is an $\mathbb{E}_2$-group whose $\mathbb{E}_2$-structure does not refine to an $\mathbb{E}_{\infty}$-structure.