The diffuse $\gamma$-ray emission from short-lived radioactive $^{26}$Al and $^{60}$Fe provides a direct probe of ongoing nucleosynthesis in the Galaxy. However, theoretical models have long struggled to reproduce the observed $^{60}$Fe/$^{26}$Al flux ratio, typically predicting values significantly higher than constraints derived from INTEGRAL/SPI observations. In this work, we investigate the impact of the recently measured, temperature-dependent stellar $\beta^-$ decay rate of $^{59}$Fe on the nucleosynthesis of these isotopes. We compute a grid of non-rotating massive star models ($14$-$80$ M$_\odot$) at solar metallicity using the MESA code, coupled with a rigorous numerical resolution analysis. We find that the updated rate significantly suppresses the net production of $^{60}$Fe by approximately 0.28 dex ($\sim 47\%$) compared to models using LMP theoretical rates, while leaving $^{26}$Al yields virtually unchanged. This reduction is primarily driven by the enhanced $\beta^-$ decay during convective carbon shell burning. Integrating these yields over a standard Salpeter Initial Mass Function, we predict a Galactic flux ratio of $\sim 0.18$, which is in excellent agreement with the observed value of $0.184 \pm 0.042$. Furthermore, this ratio exhibits a weak dependence on the IMF slope. Our results indicate that the updated nuclear physics input significantly alleviates the long-standing $^{60}$Fe overproduction problem, bringing theoretical predictions into much closer alignment with current Galactic observations.