While sightseeing aspects of Calculus of Variations, the following fact elludes me: there is a plethora of new definitions which seem redundant to me. This phenomenom happens, of course, with other subjects: for instance, one can argue that a vector space is a module over a field instead of making a "new" definition for a vector space (this is not so good of an example due to one being often introduced to vector spaces before modules, but it gets my core idea). However, when such phenomena happens in these other cases, there usually is a nice reference in the literature which makes the correspondence of definitions clear. But in all references on calculus of variations I've seen, a "variation" is a new object that is defined, and I can't see why one should not regard this as simply a case of Fréchet-derivation.

This happens even if one take a path-space of paths connecting a point $a$ to $b$, for instance. Let's consider the space $C^1([0,1], \mathbb{R}^n, a,b)$ the space of $C^1$ paths with initial point $a$ and endpoint $b$. This is an affine space over the normed vector space $C^1([0,1], \mathbb{R}^n, 0,0)$, so we have a bona-fide Fréchet-derivative, and hence we can talk about critical points. The "variations" are simply elements of the vector space.

Therefore, my questions are: Am I missing something? More precisely, is my point of view lacking or incorrect in some aspect?

If not, why isn't this approached in this way in the literature?

Sidenote: This question has been asked in MSE with an active bounty, but received an answer which I did not find sufficient for some reasons... but more objectively for the fact that it did not justify the inexistence of this point of view in the literature (or at least in the books I've seen. I'm happy to be provided a reference which does this. Even so, it seems to be rare, if it exists). I would appreciate an answer that could also address this point.