Generative adversarial networks (GANs) have recently been proposed as a potentially disruptive approach to generative design due to their remarkable ability to generate visually appealing and realistic samples. Yet, we show that the current generator-discriminator architecture inherently limits the ability of GANs as a design concept generation (DCG) tool. Specifically, we conduct a DCG study on a large-scale dataset based on a GAN architecture to advance the understanding of the performance of these generative models in generating novel and diverse samples. Our findings, derived from a series of comprehensive and objective assessments, reveal that while the traditional GAN architecture can generate realistic samples, the generated and style-mixed samples closely resemble the training dataset, exhibiting significantly low creativity. We propose a new generic architecture for DCG with GANs (DCG-GAN) that enables GAN-based generative processes to be guided by geometric conditions and criteria such as novelty, diversity and desirability. We validate the performance of the DCG-GAN model through a rigorous quantitative assessment procedure and an extensive qualitative assessment involving 89 participants. We conclude by providing several future research directions and insights for the engineering design community to realize the untapped potential of GANs for DCG.