A "nice" category $\mathcal{C}$ should be (for the purposes of this question) locally presentable at a minimum, and maybe a bit more. One might require $\mathcal{C}$ to be (in roughly order of increasing restrictiveness)

• ABn for some $n$.

• Grothendieck

• locally finitely presentable

• the category $\mathsf{QCoh}(X)$ of quasicoherent sheaves on a scheme $X$ (possibly with further adjectives)

• etc.

In particular: if $\mathsf{QCoh}(X)$ has enough projectives, then is $X$ a disjoint union of affine varieties?

Clarification: Just to be clear, I'm well aware of the Freyd-Mitchell embedding theorem. This is not a question about how close abelian categories are to module categories -- it's a question about how restrictive it is for an abelian category to have enough projectives. The local presentability hypothesis rules out the duals of categories of sheaves for instance.

Motivation / Evidence: I'm thinking of things like this result: the category of sheaves on a locally connected topological space has enough projectives iff that space is an Alexandroff space -- a very restrictive condition. I suspect that the category of sheaves on an Alexandroff space is a module category additive presheaf category. Although on reflection, the category of sheaves on an Alexandroff space need not be an additive presheaf category -- in particular, it need not be locally finitely presentable. So perhaps one should assume that $\mathcal C$ is locally finitely presentable for the purposes of this question.

For another example in this direction, consider the fact that the category of quasicoherent sheaves on a smooth projective variety of dimension >0 over a field never has enough projectives. (See my CW answer below for a more general result).

Alternative formulation: If we assume that $\mathcal C$, in addition to being a "nice" abelian category with enough projectives, has a compact generator, then is $\mathcal C$ a module category? This would follow from the formulation above, since an additive presheaf category with a compact generator is a module category; but it could conceivably be easier to show. It would also settle a form of the question about categories $\mathsf{QCoh}(X)$.