I don't really know what you need, but here is my understanding:
There are two ways to define a sheaf of abelian groups:
I. as a local homeomorphic projection with some properties on fibers.
II. as a presheaf (with S1. gluing and S2. local-global uniqueness properties).
Sheafification is a functor from II to I, which can be viewed as a natural transformation of sheaf functors. For a presheaf satisfying S1 and S2, sheafification is an equivalence. Thus one may want to identify I and II as a definition of a sheaf, but this is NOT the correct thing to do, and the obstruction to identifying them is exactly the cohomology of the subsheaf.
In detail, on one hand, the group structures in I are given fiberwise and homomorphisms of sheaves are continuous maps which are homomorphisms on the fibers; on the other hand, the group structures in II are given more globally on open sets. So passing from I to II is exactly asking the lifting question from local to global.
When we define exact sequences stalkwise, it may fail to be surjective when passing to global, for example, let X be $C^*$, the exact sequence $0 \rightarrow Z \rightarrow \mathcal{O} \rightarrow \mathcal{O^*} \rightarrow0 $, which fails to be surjective since log is not defined on $C^*$.
That means we have some global sections in the "quotient sheaf" which are not quotients of global sections; this is because some local data do glue together in the locally compatible sense, but after gluing fail to be quotients of global sections. See here for another example.
This suggests that it may fail if we want to define a quotient sheaf in the brutal way that you suggested, because you may fail to have enough global data to realize a gluing. For a sheaf functor you really mean a presheaf functor, despite it commuting with colimit, you also need to check S1 & S2 for being a sheaf.
It is worth mentioning there are also important cases of presheaves which fail S1, so we have to do sheafification in order to give sheaf cohomology. That is the presheaf of cochains. In this case we lose the clarity of supports and restriction, and this give rise to the difficulty of considering cohomology of subsets and of pairs.
