The "motivic motivation" is that by idempotent completing correspondences over a finite field one obtains a category of homological motives where Kunneth decompositions of diagonals are available. Moreover, over any field the category of numerical motives is abelian semi-simple.

The proof of the latter statement is relatively simple, and can probaly be generalized to other relevant settings. Yet I do not think that there exists any "deep" and general yoga that says that idempotent completions are crucially important.

The "motivic motivation" is that by idempotent completing correspondences over a finite field one obtains a category of homological motives where Kunneth decompositions of diagonals are available. Moreover, over any field the category of numerical motives is abelian semi-simple.

The proof of the latter statement is relatively simple, and can probaly be generalized to other relevant settings. Yet I do not think that there exists any "deep" and general yoga that says that idempotent completions are crucially important (and that is really relevant for motives).

Another observation is that over a field of positive characteristic $p$ we don't know whether Voevodsky motives of arbitrary varieties belong to the (smallest strict) triangulated subcategory generated by motives of smooth projectives, but they belong to the subcategory generated by Chow motives (if the characteristic $p$ is invertible in the coefficient ring).