An exact (small) category $P$ is an environment in which we make sense of the "put-together"-edness of objects via (short) exact sequences. It seems like the K-theory of an exact category encodes the high order relations of how objects fit together, but I can't see how the $Q$-construction is the natural medium for this.

$\bullet$ $0 \to B \to A \to C \to 0$ means that $A$ is put together in some way from $B$ and $C$. Letting $E$ denote the category of short exact sequences in $P$ (with obvious morphisms), we see that this holds in the large as well: If $p$ (resp. $q$) is the projection functor $E \to P$ that sends a s.e.s. to the first (resp. last) object in the sequence, then $(K_i(p), K_i(q)): K_i(E) \to K_i(P) \oplus K_i(P)$ is an isomorphism.

$\bullet$ Moreover, results like devissage, localisation, and resolution are also easily seen to be reassurances that we are distilling a very sensible notion of put-together-edness.

Can anyone offer a (non-circular) reason as to why the Q-construction is

right?

I don't see the sense or utility considering a category in which a morphism from $X$ to $Y$ is an isomorphism of $X$ with an (admissable) subquotient of $Y$.