Radar would probably not, in this case. At the point where it crossed the road the train was not moving toward or away from the car, so radar would have to filter it out as a stationary object (stationary objects have to be aggressively filtered with radar to avoid false positives from metal signs or stopped cars near the road). Even if it somehow was detected as a moving object, sometimes trains stop in crossings, at which point it would definitely be a stationary object and radar would be useless.
You are correct. The point is radar in cars has really bad angular resolution so it can't distinguish stuff that's directly in front of you (like a stopped train on a crossing) from stuff that's just to the side (like signs). Since both signals appear identical to the radar, and you don't want your car to emergency brake every time you go past a sign, cars filter basically ignore stationary objects detected by radar.
That isn't what you want, but it's a limit of current commercially viable radar technology.
