<p>Understanding extinction probabilities in branching processes is pivotal for epidemiology and population dynamics. Traditional models often assume a fixed generation time, resulting in extinction probabilities determined solely by offspring distributions and remaining unchanged over time. By contrast, our study incorporates the generation time into the analysis, considering how the timing of each generation influences extinction dynamics. By focusing on the finite-time risk of extinction, our approach reveals that shorter generation times can lead to a temporary increase in the risk of population or epidemic die-out. We support our findings with precise fixed-point analyses and numerical integration techniques based on real data from various infectious diseases. Although the long-term probability of extinction does not change, the transient dynamics show that a faster-paced transmission process may elevate early extinction risk. The study highlights the crucial role of transmission timing in epidemic modeling and indicates that accounting for generation time can provide new perspectives for developing effective public health strategies and outbreak control measures.</p>