When, as a student in my first weeks, I first read the definition of a group, the concept of groups looked very natural and motivated to me. But when I read the definition of a ring some time later, I had no idea what rings should be about and how they were motivated. Especially the distributive property let me wonder how much of the multiplicative structure is already determined by the addition.
All of this is has been two years ago. When I recently started to read Algebra: Chapter $\it 0$ by Paolo Aluffi, a lot of thinks I had not understood before, became clear to me. Mostly because of the use of categorical language which really works well with the way I think about mathematics.
I even came to peace with the concept of rings, when Aluffi explained them as coming up in the context of studying the homsets of abelian groups (we take the group operation as addition and the composition as multiplication).
I have been very happy about this explanation, but then, instead of studying ring by using this approach, Aluffi used the 'classical' definition of rings again.
My question is: Why?
Is it not much more promising to study rings by studying the homsets of abelian groups? I know that there is not any loose of information by just using the classical definition, but using the definition via homsets looks a lot more natural to me.
I know some category theory (at least I read Emily Rhiels book Category Theory in Context) so I could read I bit trough nLab and soon found out that a ring is equivalent to a preadditive category with only one object (encapturing pretty much what I called 'homset definiton'). So why don't we study thous preadditive categories instead? They seem to have far more 'visible structure' then rings have in their usual definition.
(I know that this is kind of a soft question. I do not look in particular for a kind of 'rigorous answer', but rather for your personal view towards this.)