What is wrong with the following argument about splitting an exact sequence of abelian groups?
Thank you @FiMePr for a hint that eventually cleared everything up. Indeed, generically a map $B\to A\oplus C$ doesn't exist. The propositions (a), (b) allow for the explicit construction of maps $B\to A\oplus C$ and $A\oplus C\to B$, respectively. It is then easy to check that these maps make the squares of which they are a part of commutative. One then invokes the five lemma to prove that the maps are in fact isomorphisms, completing the proof.