We study spin-lattice relaxation times of electron spins in Er${}^{3+}$:CaWO${}_4$ coupled to a low-mode volume superconducting resonator at millikelvin temperatures in a regime where resonant phonons are cooled to near their quantum ground state. At large magnetic field supporting strong phonon-emission rates, we observe a noticeable increase in relaxation times with increasing spin spectral density $c/{\Gamma }_{\mathrm{\text{inh}}}$, which exhibits a $[tanh(\hslash {\omega }_0/2k_{\mathrm{\text{B}}}T){]}^2$ contribution. These observations are typical of a phonon-bottlenecked spin relaxation, and have implications for quantum technologies that exploit rare-earth spin ensembles as coherent resources.