How do we choose when confronted with many alternatives (e.g., choosing which soda to buy at the supermarket)? While studies of the decision mechanisms underlying two- to three-alternative choice are common in the literature, there is comparably little decision modeling work with larger choice sets, despite their prevalence in everyday life. Even further, there is an apparent disconnect between research in small choice sets, supporting a process of gaze-driven evidence accumulation, and research in larger choice sets, arguing for models of optimal choice, satisficing, and hybrids of the two. Here, we bridge this divide by comparing these models in a many-alternative value-based choice experiment with 9, 16, 25 or 36 choice alternatives. We find that human subjects' choice behaviour does not match the assumptions of satisficing or optimal choice, while a hybrid of the two captures response times and choices well. Yet, only a process of gaze-driven evidence accumulation is able to also accurately capture the observed association between gaze allocation and choices.