This paper presents a fast, learning-free registration method tailored to clinical constraints. While shape registration is fundamental to medical imaging, existing techniques often force a trade-off between computational scalability and strict topology preservation. Current diffeomorphic methods are either computationally expensive or demand extensive pre-processing, such as training neural networks on large datasets. This renders them impractical for clinical studies, which often rely on limited cohorts. Image-based methods also require large memory banks to process high-resolution data, while surface-based approaches have received less attention in the deep learning era. In this context, we propose a principled registration algorithm that alternates between a feature-aware matching scheme, a conjugate gradient solver and a diffeomorphic shooting step. Our pipeline can handle point clouds, curves and surface meshes. It is compatible with clinical workflows and can register a pair of shapes sampled with 10k points each in one second, without landmarks, pre-alignment, or supervision. We demonstrate consistent results on several anatomical structures and showcase the efficiency of our method by computing an atlas from hundreds of complex vertebral scans in a few minutes. Upon publication in September 2026, our code will be released under the permissive MIT license at: scikit-shapes.github.io.