We report on efficient spin injection in p-doped $(\mathrm{In},\mathrm{Ga})\mathrm{As}/\mathrm{Ga}\mathrm{As}$ quantum-dot (QD) spin light-emitting diodes (spin LEDs) under zero applied magnetic field. A high degree of electroluminescence circular polarization (${P}_{c}$) \ensuremath{\sim}19% is measured in remanence up to 100 K. This result is obtained thanks to the combination of a perpendicularly magnetized $\mathrm{Co}\text{\ensuremath{-}}\mathrm{Fe}\text{\ensuremath{-}}\mathrm{B}/\mathrm{Mg}\mathrm{O}$ spin injector allowing efficient spin injection and an appropriate p-doped $(\mathrm{In},\mathrm{Ga})\mathrm{As}/\mathrm{Ga}\mathrm{As}$ QD layer in the active region. By analyzing the bias and temperature dependence of the electroluminescence circular polarization, we evidence a two-step spin-relaxation process. The first step occurs when electrons tunnel through the $\mathrm{Mg}\mathrm{O}$ barrier and travel across the $\mathrm{Ga}\mathrm{As}$ depletion layer. The spin relaxation is dominated by the Dyakonov-Perel mechanism related to the kinetic energy of electrons, which is characterized by a bias-dependent ${P}_{c}$. The second step occurs when electrons are captured into QDs prior to their radiative recombination with holes. The temperature dependence of ${P}_{c}$ reflects the temperature-induced modification of the QD doping, together with the variation of the ratio between the charge-carrier lifetime and the spin-relaxation time inside the QDs. The understanding of these spin-relaxation mechanisms is essential to improve the performance of spin LEDs for future spin optoelectronic applications at room temperature under zero applied magnetic field.