For cells to function, the concentrations of all proteins in the cell must be maintained at the proper levels (proteostasis). This task--complicated by cellular stresses, protein misfolding, aggregation, and degradation--is performed by a collection of chaperones that alter the configurational landscape of a given client protein through the formation of protein-chaperone complexes. The set of all such complexes and the transitions between them form the proteostasis network. Recently, a computational model was introduced (FoldEco) that synthesizes experimental data into a system-wide description of the proteostasis network of E. coli. This model describes the concentrations over time of all the species in the system, which include different conformations of the client protein, as well as protein-chaperone complexes. We apply to this model a recently developed analysis tool to calculate mediation probabilities in complex networks. This allows us to determine the probability that a given chaperone system is used to mediate transitions between client protein conformations, such as folding, or the correction of misfolded conformations. We determine how these probabilities change both across different proteins, as well as with system parameters, such as the synthesis rate, and in each case reveal in detail which factors control the usage of one chaperone system over another. We find that the different chaperone systems do not operate orthogonally and can compensate for each other when one system is disabled or overworked, and that this can complicate the analysis of "knockout" experiments, where the concentration of native protein is compared both with and without the presence of a given chaperone system. This study also gives a general recipe for conducting a transition-path-based analysis on a network of coupled chemical reactions, which can be useful in other types of networks as well.