I am trying to understand a big-picture for Algebraic Geometry:

Given a category of commutative rings $\mathrm{CRing}$, we can create objects that locally look like these objects called schemes. This is done by taking the topos of sheaves of $\mathrm{CRing}$ with respect to a Zariski (or even etale) topology. The subcategory which has objects covered by representables (objects mapped from $\mathrm{CRing}$ via yoneda embedding and sheafification) via open maps.

So this overall constructs some "locally isomorphic to rings" object. There are a few issues one needs to consider which can be fixed by restricting to finitely presentable rings (does this make the schemes (quasi-)compact?).

So in analogy, if you want to study $\mathrm{CRing}$ you can study modules instead which are "generalised functions".

So we can take our previous schemes and generalise them even more by thinking about sheaves over them with respect to another topology, this time we have all sorts of choices (fppf, zar, h-, etale etc.).

Now these form a "big zariski topos" so we have an internal language, thus we can describe rings and modules internally. Categories of these become sheaves of rings and (quasi-)coherent sheaves respectively.

The category of (quasi-)coherent sheaves over some scheme can have its simplicial objects studied. Traditionally this is done via the Dold-Kan correspondence and gives the derived category construction. This homotopy category is a big package describing (co)homology theories. Other cohomology theories that arise in AG can be studied by changing the topology chosen earlier.

Now this big picture has some quite broad strokes and many different aspects which one can generalise. Why stick with with rings when you can use $E_\infty$-rings, why consider sheaves when you can consider $\infty$-stacks etc.

So my question is: How accurate is this picture?

I know I am being a bit cheeky by calling all of this algebraic geometry when there is so much more, (for example classifying varieties birationally etc).

Oh and just for note: cohomologies I have described above are worth studying, not because of all this silly category theory but because they describe and classify important and useful invariants of the geometric objects in question.