Ho(C) is badly behaved
The homotopy category of an $(\infty,1)$-category is not well behaved in the categorical sense. This is perhaps believable because the homotopy category truncates all higher morphisms. These failures include:
In fact this is a reason for developing and working with model categories: when we pass to fibrant/cofibrant resolutions, limits of these special class of objects do align with homotopy limits.
Another moral reason why the homotopy category is insufficient is because it is an ordinary localization corresponding to the reflective subcategory of small categories in simplicial sets. But the setting we wish to work in is that of an $(\infty,1)$-category, so we actually want to consider the $(\infty,1)$-reflective subcategory and its corresponding localization.
The abstract construction above is modelled by Bousfield localization.
References
- https://math.stackexchange.com/questions/3795052/what-is-an-infinity-category-really (particularly shibai’s answer)