Site-controlled pyramidal quantum dots (PQDs) are high-symmetry structures grown on (111) B-oriented substrates by metal-organic vapor phase epitaxy. Given the relatively straightforward technique used for their growth and the high degree of dot uniformity achievable, they are natural candidates for the realization of reliable single and entangled photon sources. Nevertheless, even though (111)-oriented surfaces show, in terms of lattice symmetry, ideal characteristics for the production of entangled photon pairs, in PQDs the fine-structure splitting (FSS) can vary significantly depending on the dot composition and growth conditions. Here, we present the results of a study of the FSS in six samples of different composition. We investigate correlations among FSS and other parameters, such as biexciton binding energy and exciton lifetime, and extract information on how to manipulate the dot to reproducibly obtain a nearly zero FSS. To gain more detailed insight into the electronic properties of (111)-oriented InxGa1-xAs PQDs, we also perform calculations based on an eight-band k.p model. These calculations are in good qualitative agreement with the experimental outcomes and outline the importance of piezoelectric potentials and of the carrier confining potential for predicting the optoelectronic properties.