Atlas construction is fundamental to medical image analysis, offering a standardized spatial reference for tasks such as population-level anatomical modeling. While data-driven registration methods have recently shown promise in pairwise settings, their reliance on large training datasets, limited generalizability, and lack of true inference phases in groupwise contexts hinder their practical use. In contrast, model-driven methods offer training-free, theoretically grounded, and data-efficient alternatives, though they often face scalability and optimization challenges when applied to large 3D datasets. In this work, we introduce DARC (Diffeomorphic Atlas Registration via Coordinate descent), a novel model-driven groupwise registration framework for atlas construction. DARC supports a broad range of image dissimilarity metrics and efficiently handles arbitrary numbers of 3D images without incurring GPU memory issues. Through a coordinate descent strategy and a centrality-enforcing activation function, DARC produces unbiased, diffeomorphic atlases with high anatomical fidelity. Beyond atlas construction, we demonstrate two key applications: (1) One-shot segmentation, where labels annotated only on the atlas are propagated to subjects via inverse deformations, outperforming state-of-the-art few-shot methods; and (2) shape synthesis, where new anatomical variants are generated by warping the atlas mesh using synthesized diffeomorphic deformation fields. Overall, DARC offers a flexible, generalizable, and resource-efficient framework for atlas construction and applications.