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 arises from the fact that on the side of Kan complexes, there are some elements 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.

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.