One viewpoint goes as follows: the 2-categorical structure on groups can be seen as coming from inner automorphisms, so that a 2-cell is given by an inner automorphism that translates one map to the other. Now, inner automorphisms of an object can be defined in any category (see e.g. this paper) using the notion of isotropy group, a particular functor from the starting category to groups.

Moreover, one can use these abstract inner automorphisms to promote any category into a 2-category (albeit not in a functorial way: a given functor might not become a 2-functor). I have some results on how 2-categorical limits and colimits behave in the resulting 2-category. In particular, as soon as your starting category is finitely cocomplete, it has all coinserters iff the isotropy functor is representable. Moreover, all limits and connected colimits of the underlying category satisfy the required two-dimensional universal property as well. However, inserters, equifiers or 2-d coproducts exist iff the isotropy is trivial. For what it's worth, I've given a talk about this stuff here but there's no publicly available writeup yet.

One can debate if this counts as a "deep reason", as ultimately the general proof that strict inserters do not exist boils down to the kind of situation you considered. That said, this does put the observation in context, so perhaps it still counts.

One viewpoint goes as follows: the 2-categorical structure on groups can be seen as coming from inner automorphisms, so that a 2-cell is given by an inner automorphism that translates one map to the other. Now, inner automorphisms of an object can be defined in any category (see e.g. this paper) using the notion of isotropy group, a particular functor from the starting category to groups.

Moreover, one can use these abstract inner automorphisms to promote any category into a 2-category (albeit not in a functorial way: a given functor might not become a 2-functor). Me and Pieter Hofstra have some results on how 2-categorical limits and colimits behave in the resulting 2-category. In particular, as soon as your starting category is finitely cocomplete, it has all coinserters iff the isotropy functor is representable. Moreover, all limits and connected colimits of the underlying category satisfy the required two-dimensional universal property as well. However, inserters, equifiers or 2-d coproducts exist iff the isotropy is trivial. For what it's worth, I've given a talk about this stuff here but there's no publicly available writeup yet.

One can debate if this counts as a "deep reason", as ultimately the general proof that strict inserters do not exist boils down to the kind of situation you considered. That said, this does put the observation in context, so perhaps it still counts.