Natural disasters demand rapid damage assessment to guide humanitarian response. Here, we investigate whether medium-resolution Earth observation images from the Copernicus program can support building damage assessment, complementing very-high resolution imagery with often limited availability. We introduce xBD-S12, a dataset of 10,315 pre- and post-disaster image pairs from both Sentinel-1 and Sentinel-2, spatially and temporally aligned with the established xBD benchmark. In a series of experiments, we demonstrate that building damage can be detected and mapped rather well in many disaster scenarios, despite the moderate 10m ground sampling distance. We also find that, for damage mapping at that resolution, architectural sophistication does not seem to bring much advantage: more complex model architectures tend to struggle with generalization to unseen disasters, and geospatial foundation models bring little practical benefit. Our results suggest that Copernicus images are a viable data source for rapid, wide-area damage assessment and could play an important role alongside VHR imagery. We release the xBD-S12 dataset, code, and trained models to support further research at <code>https://github.com/prs-eth/xbd-s12</code>.