Generative AI (GenAI) models excel in their ability to recognize patterns in existing data and generate new and unexpected content. Recent advances have motivated applications of GenAI tools (e.g., Stable Diffusion, ChatGPT) to professional practice across industries, including product design. While these generative capabilities may seem enticing on the surface, certain barriers limit their practical application for real-world use in industry settings. In this position paper, we articulate and situate these barriers within two phases of the product design process, namely "getting the right design" and "getting the design right," and propose a research agenda to stimulate discussions around opportunities for realizing the full potential of GenAI tools in product design.