The phase space modification associated with a nonvanishing effective mass for the primary gluons, ${M}_{g}=0.66\ifmmode\pm\else\textpm\fi{}0.08$ GeV for the $\frac{J}{\ensuremath{\psi}}$ and ${M}_{g}=1.17\ifmmode\pm\else\textpm\fi{}0.08$ GeV for the $\ensuremath{\Upsilon}$, is shown to be crucial for a consistent description of the photon spectrum from their radiative decays and for the determination of ${\ensuremath{\alpha}}_{s}$ from the recent, precise quarkonia decay branching ratios. In this approach, the role of the relativistic corrections is marginal and, after applying the ${M}_{g}$-dependent corrections, a good agreement is obtained with the relative perturbative running, ${\ensuremath{\alpha}}_{s}({m}_{c})\ensuremath{\sim}0.30\ifmmode\pm\else\textpm\fi{}0.02$ and ${\ensuremath{\alpha}}_{s}({m}_{b})\ensuremath{\sim}0.21\ifmmode\pm\else\textpm\fi{}0.01$, and with the extrapolation from deep inelastic scattering. On the other hand, for ${M}_{g}=0$, the analysis of all experimental $c\overline{c}$ and $b\overline{b}$ quarkonia branching ratios is consistent with the same effective value of the strong interaction coupling constant ${\ensuremath{\alpha}}_{s}^{\mathrm{eff}}\ensuremath{\sim}0.185\ifmmode\pm\else\textpm\fi{}0.010$. By assuming a "genuine," i.e., process-independent, gluon mass (\ensuremath{\sim} 1.2 GeV or higher) to be dynamically generated one predicts a strong suppression of the gluon splitting process at the $\frac{J}{\ensuremath{\psi}}$ and the hadronic final states should be mainly produced via gluon fusion into light $q\overline{q}$ pairs thus effectively reducing the fitted value of ${M}_{g}$ from the photon spectrum in $\frac{J}{\ensuremath{\psi}}\ensuremath{\rightarrow}\ensuremath{\gamma}+X$. The gluon fusion mechanism allows us to explain the structure of the hadronic final states observed in $\frac{J}{\ensuremath{\psi}}$ decays and their close similarity to the continuum ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}$ hadrons annihilation at comparable center of mass energies.