Why must one sheafify quotients of sheaves?

For presheaves (of sets or groups) we know what this particular (or any) colimit operation is: apply the operation objectiwise (for each $U$). Now the sheafification preserves colimits, hence we apply sheafification to a colimit cocone on presheaves to obtain a colimit cocone in sheaves. Doing sheafification to the presheaves which are alerady sheaves does nothing to them, but it, by the right exactness, does the correct thing to the colimit. This proves that the sheafification following the colimit in presheaves is the correct way to compute the colimit, and in that we did use the right exactness of the sheafification essentially. The fact that it is necessary does not follow from the general nonsense as there are both examples where we accidentally do not need a sheafification step and those where we do need. For the limit constructions on sheaves we never need the sheafification because the embedding of the sheaves into presheaves is left exact hence we can simply compute the limits in presheaves. It seems you had somehow an opposite impression.

You ask: "Why must one sheafify the presheaf $P(\mathcal{F}/\mathcal{G})$ then?". The answer is: "Because it is not a sheaf!" Here is an example.

Let $X=\mathbb P^1_k, \mathcal F = \mathcal O, \mathcal G=\mathcal O (-2.O)$, where $O$ denotes the origin $O=(0:1)$ . Define $\infty=(1:0)$ and let $z$ be the coordinate on $\mathbb P^1\setminus \infty$ .

Consider the covering of X by the open subsets $U_0=X\setminus \infty$ and $U_\infty =X\setminus O$. Let me denote the presheaf $P(\mathcal{F}/\mathcal{G})$ just by $P$.

Then $class(z)\in P(U_0)$ and $class(0)\in P(U_\infty)$ are sections of $P$ over the two open sets $U_0$ and $U_\infty$ of our covering which coincide on their intersection $U_{0\infty}$ for the excellent reason that $P(U_{0\infty})=0$ ! [Actually $P(U)=0$ for any open subset $U\subset X$ not containing $O$]

But these compatible sections cannot be glued to a global section of $P$ on $X$. Indeed a section of $P$ on $X$ is just a constant $c\in k$ , since $\mathcal O(X)=k$ and $\mathcal O(-2.O)(X)=0$. But that constant $c$ cannot be the glued global section , because its restriction to $P(U_0)$ is $ class(c) $ and in $P(U_0)$ we have $class(c) \neq class(z)$ since $z-c$ doesn't vanish with order two at $O$.