Can Homological algebra be done in Rmod without loss of generality? The theory and standard constructions of Homological algebra can be done at the generality of any abelian category $\mathcal{A}$, and in particular the category of $R$ modules over some ring $R$. Due to the Mitchel embedding theorem every small abelian category can be embedded into $R$ mod for some $R$ via an exact functor. This allows us to assume without loss of generality that $\mathcal{A}$ is actually just the category of $R$ modules (at least when $\mathcal{A}$ is small). If $\mathcal{A}$ is not small, then you could always just restrict to a subcategory which contains all relevant diagrams and work there.
My question is, are there any issues with doing this in general? Another way to phrase my question is, can homological algebra be done in the category of $R$ modules without loss of generality, or are there any
results in homological algebra which hold in $R$ mod that do not hold in certain large abelian categories?
 A: No, this is not sufficient. Here are some reasons which come to my mind spontaneously, and I am sure that there are lots of others, which convinced the mathematicians in the 50s and 60s to adopt the notion of an abelian category:

*

*The embedding only shows that $\mathcal{A}$ is equivalent to a full subcategory of a category of modules, but not equivalent to the whole category. So we will have to work with subcategories instead. But then we need to ask the question: why do we really care about the ambient category of modules?


*The embedding only works for small categories, and the workaround you suggest is not always possible, for example when we want to work with cocomplete abelian categories, in particular Grothendieck categories. Just think of prominent examples such as $\mathrm{Sh}(X)$ for a topological space $X$.


*Even though it might be possible in some circumstances, it is awkward and a technical complication to be dependent on a "generic model" for the theory. This is also because many natural examples do not arise in this way, and general constructions such as quotient abelian categories are not compatible with it either. (The same goes for the Whitney embedding theorem, which does not imply that manifold theory is equal to the theory of submanifolds of $\mathbb{R}^n$.)


*The Mitchell embedding theorem does not preserve / reflect everything what we need. You might get the impression that projective and injective resolutions always exist, since this holds for $R$-modules, but it absolutely fails for generic abelian categories (again, this is connected to the first point of the list). For some properties it is even unclear if we can preserve them, see MO/32173.


*The embedding theorem only tells us something about the objects of our theory, nothing about the morphisms (exact functors or variants thereof).
That being said, my impression is that the embedding theorem is "overrated" and has been used for purposes which can be solved by other means in a much more simple and elegant way. For instance, diagram chases can be carried out with a generalized notion of an element as explained in MacLane's book.
