This paper investigates the potential impact of deep generative models on the work of creative professionals. We argue that current generative modeling tools lack critical features that would make them useful creativity support tools, and introduce our own tool, generative.fashion, which was designed with theoretical principles of design space exploration in mind. Through qualitative studies with fashion design apprentices, we demonstrate how generative.fashion supported both divergent and convergent thinking, and compare it with a state-of-the-art text-based interface using Stable Diffusion. In general, the apprentices preferred generative.fashion, citing the features explicitly designed to support ideation. In two follow-up studies, we provide quantitative results that support and expand on these insights. We conclude that text-only prompts in existing models restrict creative exploration, especially for novices. Our work demonstrates that interfaces which are theoretically aligned with principles of design space exploration are essential for unlocking the full creative potential of generative AI.