Automotive radars struggle to detect objects which are not moving relative to the ground in the range direction. Such detections must be filtered aggressively to avoid false positives from stationary objects such as metal signs near or above the road.
At the point where it crossed the road the train was moving sideways relative to the car. In the range direction it was close to stationary relative to the ground (and the crossing arm was entirely stationary, of course). So I think radar would be unlikely to save this case. If the radar was looking to the side, so it could see the train cars approaching at high speed and predict that they would enter the road, then maybe. But this seems like it would also potentially have a lot of false positives e.g. at overpasses. Perhaps for that reason, I believe automotive radars like the ones Teslas used to have generally only point toward the front. And if the train had been stopped in the crossing, as they sometimes are, the radar would be unhelpful no matter where it was pointed.