A recently proposed strictly phenomenological static quark-antiquark potential belonging to the generality V(r) = -Ar-alpha + kappa r(beta) + V-0 is tested with heavy quarkonia in the context of the shifted large N-expansion method. This nonrelativistic potential model fits the spin-averaged mass spectra of the c (c) over bar, b (b) over bar and c (b) over bar quarkonia within a few MeV and also the five experimentally known leptonic decay widths of the cc and bb vector states. Further, we compute the hyperfine splittings of the bottomonium spectrum as well as the fine and hyperfine splittings of the charmonium spectrum. We give predictions for not yet observed B-c splittings. The model is then used to predict the masses of the remaining quarkonia and the leptonic decay widths of the two pseudoscalar c (b) over bar states. Our results are compared with other models to gauge the reliability of the predictions and point out differences.