Well, from what my parents have told me about their trip to New York aboard an Amtrak train, several problems arose from checking passports at Rouses Point. For example, if someone doesn't have their papers in order, they will have to get off the train and the whole train will wait for them to come back once the checkup is completely done and they're allowed to enter the country.
If this was done BEFORE boarding, it would prevent this kind of situation since everybody would already be cleared to enter the USA and there wouldn't be any nasty surprises on the way.