I only have studied basic topology, which means i haven't studied any about differentiable manifolds. I just skimmed pages on wikipedia.
Here is a simple illustration on a basic situation.
Let $E$ be a subset of $\mathbb{K}$ and $(V,\| \cdot \|)$ be a normed space over $\mathbb{K}$ and $f:E\rightarrow V$ be a function.
It is only neccessary to check whether a point $x$ of $E$ is a limit point of $E$ to make a sense to say that "$f$ is differentiable at $x$.
So that we can define the differentiation of a function whose domain is such as a closed disk in $\mathbb{C}$ or the Cantor set.
For this reason, i guess the concept of differentiable manifold does not capture all the properties from which the concept of differentiation comes (i.e. $\mathbb{K}$), since the theory of differentiable manifolds only consider open sets.
I guess that if we consider all these cases together, then may the theory look messy. So i think all the concepts of differentiable manifolds are indeed substitutes to this broad consideration.
Am i thinking correctly or am i actually not capturing the intrinsic concept of differentiation?
While I feel the generalization of integration to a locally compact Hausdorff space is very natural, i kind of feel that the theory of abstract differentiation is a bit artificial.
I think that you are missing an important point about differentiation (but don't feel ashamed, it's not an obvious point at all if you don't have enough background). The important thing here is that differentiation is a local concept, i.e. we have the concept of differentiation at a point, and this depends only on a(n arbitrarily small) open neighborhood of said point. Even at a limit point a function is differentiable if, and only if it can be extended to a differentiable function on a small neighborhood of that point. This is the reason why, when defining smooth manifold (but also topological manifolds and other similar concepts) we need only a definition which relies on local concepts, and thus bother only with (small) open sets.
A more advanced point of view is given by the concept of sheafs, but you'll probably need to build up a bit more basic knowledge to fully understand the concept.