Consider the category $\mathsf{NormVect}_1$ of normed vector spaces with short linear maps$^{\dagger}$ and the full subcategory $\mathsf{Ban}_1$ of Banach spaces with short linear maps. Both categories are complete, cocomplete, and have a closed symmetric monoidal structure, given by the projective tensor product (see here). The forgetful functor $\mathsf{Ban}_1 \to \mathsf{NormVect}_1$ is continuous (but not cocontinuous), in fact has a left adjoint (which is symmetric monoidal), the Cauchy completion.
Question. Can you name a categorical property of $\mathsf{Ban}_1$ which is useful in practice, but which is not satisfied by $\mathsf{NormVect}_1$?
Background: There is a branch called categorical Banach space theory, and I really wonder why there one does not consider the larger category of all normed vector spaces somehow as a first approximation. In functional analysis it is well-known that (and why) Banach spaces are more useful than normed vector spaces. I would like to know if or why this is also true for the corresponding categories.
$^{\dagger}$ Notice that the subscript $1$ indicates that we restrict ourselves to short linear maps, which is quite important for having the mentioned categorical properties. For me, the moral of this choice is that if you use continuous linear maps, you don't take the whole structure of the objects into account, which tends to be bad.
The category of Banach spaces is locally $\aleph_1$-presentable.