sheaf theory - Heuristic explanation of why we lose projectives in sheaves.

We know that presheaves of any category have enough projectives and that sheaves do not, why is this, and how does it effect our thinking?

This question was asked(and I found it very helpful) but I was hoping to get a better understanding of why.

I was thinking about the following construction(given during a course);

given an affine cover, we normally study the quasi-coherent sheaves, but in fact we could study the presheaves in the following sense:

Given an affine cover of X,

$Ker_2left(piright)rightrightarrows^{p_1}_{p_2} Urightarrow X$

then we can define $X_1:=Cokleft(p_1,p_2right)$, a presheaf, to obtain refinements in presheaves where we have enough projectives and the quasi-coherent sheaves coincide. Specifically, if $X_1xrightarrow{varphi}X$ for a scheme $X$, s.t. $mathcal{S}left(varphiright)in Isom$ for $mathcal{S}(-)$ is the sheaffication functor, then for all affine covers $U_ixrightarrow{u_i}X$ there exists a refinement $V_{ij}xrightarrow{u_{ij}}U_i$ which factors through $varphi$.

This hinges on the fact that $V_{ij}$ is representable and thus projective, a result of the fact that we are working with presheaves. In sheaves, we would lose these refinements. Additionally, these presheaves do not depend on the specific topology(at the cost of gluing).

In this setting, we lose projectives because we are applying the localization functor which is not exact(only right exact). However, I don't really understand this reason, and would like a more general answer.

A related appearance of this loss is in homological algebra. Sheaves do not have enough projectives, so we cannot always get projective resolutions. They do have injective resolutions, and this is related to the use of cohomology of sheaves rather than homology of sheaves. In paticular, in Rotman's Homological Algebra pg 314, he gives a footnote;

In The Theory of Sheaves, Swan writes "...if the base space X is not discrete, I know
of no examples of projective sheaves except the zero sheaf." In Bredon, Sheaf Theory:
on locally connected Hausdorff spaces without isolated points, the only
projective sheaf is 0

addressing this situation.

In essence, my question is for a
heuristic or geometric explanation of
why we lose projectives when we pass
from presheaves to sheaves.

Thanks in advance!