As noted in Tag 01WZ of Kerodon, the morphism spaces of an $(\infty,2)$-category $C$ can fail to be $\infty$-categories, in contrast to the pinched morphism spaces of $C$.

How should one think about this fact, intuitively? Say, is there some illuminating example illustrating what goes wrong for morphism spaces compared to pinched morphism spaces?

Second, I'm also wondering about whether this problem might be "rectifiable" in the following sense: is there an appropriate notion of "weak equivalence" of $(\infty,2)$-categories making the following statement true?

Given any $(\infty,2)$-category $C$, there exists an $(\infty,2)$-category $C'$ such that

1. The morphism spaces of $C'$ are $\infty$-categories;
2. The $(\infty,2)$-categories $C$ and $C'$ are weakly equivalent.