Localization effects are central to disordered electronics and photonics. In electronics, Anderson localization governs electron confinement in randomly perturbed lattices. Similarly, its photonic counterpart inhibits light transport via disorder – but with a unique exception: Brewster-anomaly delocalization, where the Brewster effect prevents multiple-scattering interference and counteracts the localization. Despite extensive research in electronics and photonics separately, the intricate role of localization effects in free-electron–light interactions – vital for lasers, accelerators, microscopy and spectroscopy, and quantum information – remains largely unexplored. At the same time, localization effects are widely regarded as a key factor limiting the efficient coupling between free electrons and light in random media. Here we overcome this key limitation via the unconventional interplay between Brewster-anomaly delocalization and free-electron radiation. In this way, free-electron radiation can be localization-free, intense and directional even in strongly disordered, unengineered multilayers. Essentially, this delocalization-mediated free-electron radiation is remarkably invariant not only to the random-medium configuration, but also to the light frequency and the electron velocity. Our findings unlock new opportunities for particle detectors and achromatic light sources operating in easy-to-fabricate complex media at previously inaccessible frequencies.