Brain-body co-optimization remains a challenging problem, despite increasing interest from the community in recent years. To understand and overcome the challenges, we propose exhaustively mapping a morphology-fitness landscape to study it. To this end, we train controllers for each feasible morphology in a design space of 1,305,840 distinct morphologies, constrained by a computational budget. First, we show that this design space constitutes a good model for studying the brain-body co-optimization problem, and our attempt to exhaustively map it roughly captures the landscape. We then proceed to analyze how evolutionary brain-body co-optimization algorithms work in this design space. The complete knowledge of the morphology-fitness landscape facilitates a better understanding of the results of evolutionary brain-body co-optimization algorithms and how they unfold over evolutionary time in the morphology space. This investigation shows that the experimented algorithms cannot consistently find near-optimal solutions. The search, at times, gets stuck on morphologies that are sometimes one mutation away from better morphologies, and the algorithms cannot efficiently track the fitness gradient in the morphology-fitness landscape. We provide evidence that experimented algorithms regularly undervalue the fitness of individuals with newly mutated bodies and, as a result, eliminate promising morphologies throughout evolution. Our work provides the most concrete demonstration of the challenges of evolutionary brain-body co-optimization. Our findings ground the trends in the literature and provide valuable insights for future work.