In Europe the view is that bigger vehicle = bigger responsibility. In the US the attitude seems to be bigger vehicle = bigger right of way. The party who stands to lose the most is expected to yield the right of way.
Are trains required to stop at level crossings in Europe, because they have the bigger responsibility?
No. There's a big song and dance here (one country) every time someone is foolish enough to cross when the barrier is down. So much that the govt and rail network want to eliminate them when it is just a handful of people a year. Now road vehicles getting hit is dangerous to train and track but I am talking about pedestrians.
The railway's responsibility is having a ten-nines reliable (or whatever it is) signalling system, to be practically certain that the lights, alarms and barriers of the level crossing work correctly every time. (Or fail safe, i.e. fail to the barriers lowering.)