The application of Bayesian Optimization using Gaussian Processes (BO-GP) for global optimization problems is ubiquitous across scientific disciplines because, beyond good performance, it supports exact inference, interpretability, and straightforward uncertainty quantification. In this paper, we revisit the biological application of BO-GP in searching trait spaces for genomic prediction, which uses genome-wide marker information to predict breeding values for agronomically important traits. Genomic predictions help breeders select desirable plants earlier in the field season without waiting to observe traits later. While these search spaces are known to be sharp and aperiodic, BO-GP is considered a feasible approach. However, our work finds that a simple random search surprisingly achieves comparable performance to BO-GP while requiring significantly less computing cost. Through a careful investigation, we can explain this observation as a fundamental limitation of BO-GP for sharp and aperiodic functions -- where the incompatible structure results in samples similar to random search but with higher computational cost. Our results highlight a blind spot in the current use of BO-GP for scientific applications, such as trait prediction, with sharp and aperiodic search spaces.