homotopy theory – construction of $K_0$-group and Karoubian completion

Let $A$ be a ring. The $K_0$ group of $A$ can be defined in most
old fashioned way as the Grothendieck group of the set of isomorphism classes
of its finitely generated projective $R$ modules, regarded as a monoid under direct sum.

One can show that this $K_0(R)$ coincides with Quillen-construction
$K_0(P_A) := pi_1(BQ(P_A),0)$ if $P_A$ is the category of finitely generated projective A-modules. The important issue of Q-construction is that it associates $K_i$ groups to every
exact category. In more general setting applying S-construction
one can associate $K$-theory groups to categories with cofibrations (called
Waldhausen categories).

If we go back to the first construction of $K_0(R)$ we observe that the
auxilary category $P_A$ of finitely generated projective A-modules is formally
the Karoubi or preudo-abelian completion of the category $F_A$ of finitely generated free $A$-modules.
What we see is that the preudo-abelian completion or preudo-abelian categories play
somehow a central role for the definition of algebraic $K_0(R)$ group.

My question is how concretely is the pseudo-abelian completion involved in the
construction of $K_0$ group in more general setting like in Quillen’s construction
or S-construction if we start with an arbitrary exact or Waldhausen category $mathcal{C}$.

One could also ask if this definition of $K_0(A)$ in algebraic $K$-theory
involving pseudo-abelian completion of fin.gen. free $A$-modules is something
special what happens only in algebraic (and topological) $K$-theory or
is the pseudo-abelian completion here based on a more general principle
used in construction of $K_0$ in $K$-theories in general setting?

The question is closely related to this discussion that mostly uncovers the
obstructional relation between existence of nonzero negative $K$-groups
Karoubian completeness.
Now I would like to understand if and how the pseudo-abelian completion is
involved in construction of $K_0$.