It is well know that Karoubian categories (also called idempotent-complete categories) are living between additive and Abelian categories. While one of the most famous advantages to work with abelian cetegories is that they are closed under building kernels and cokernels of arbitrary morphism, the Karoubian categories have a slightly weaker property that only idempotent morphisms (i.e. $p$ with $p^2=p$) from there share this property.
Nevertheless it seems that in diverse constructions Karoubian categories or Karoubian envelopes of additive categories provide a more natural setting than Abelian categories.
I hope my question not becomes too broad: What is the philosophical meaning behind Karoubian categories or say in simpler words why in some constructions they are more prefered (e.g. category of pure motives and in K-theory) in contrast to say at first glance more 'flexible' abelian categories?
My natural guess is that if we think about the construction
of of the category of pure motives we start with the category
of smooth varieties over a base field and pass after
application of this magic Karoubian completion functor to
idempotent-complete categories. Since it's not abelian it contains by definition only kernel and cokernels of idempotent morphisms but that's all we need there to proceed the constrution.
This lead me to conjecture that the main advantage of Karoubian categories in contrast to Abelian categories mights show when one have to perform a construction where one starts with a certain preadditive category but the construction requireres a category having at least some kernels and cokernels.
Now one can pass canonically to the Karoubian completion or extend the initial category to an Abelian category. But exactly here I see an obstacle with the secound and at first glance more 'natural' approach:
Does there always a way to embedd a preadditive category in an abelian category? If yes, seemingly the disadvantage of this approach seems to be that this Abelian category is much harder to control, while the Karoubian completion is constructed quite canonically and behaves more 'similar' to initial preadditive category.
Questions: Is what I tried to sketch above exactly the motivation why Karoubian categories are in some constructions more prefered then Abelian categories?
Are there more reasons making Karoubian categories 'interesting'? Is there any intuition or important example one should have in mind how to think about Karoubian categories?