Short answer: I would recommend taking it. It's mandatory for you to carry it in the Netherlands, mostly mandatory in Germany and Switzerland. Depending on your tickets, train guards in Germany may also want to see some ID (technically, it might be required with an online ticket all over Europe but I never saw anyone care outside of Germany).
Police checks are not unheard of and I wouldn't expect much flexibility from the police, especially in those three countries. If there is a check, I would expect the most likely outcome to be a lecture or a fine but I would not rule out being forced to get off the train and back to your point of origin in the next one or some detention while they figure out what to do with you. Of course, it's also entirely possible that you won't have to show your passport to anyone at all but is it worth the risk of a ruined holiday?
Your US driving license legally doesn't prove anything. It doesn't establish your citizenship, it doesn't prove you have a visa and therefore your status in the Schengen area, and it does not meet the standard requirements for police checks (Germany and the Netherlands in particular have laws that specify exactly what documentation is expected). It might still be useful as it does provide some evidence of who you are and also that you are genuinely a tourist coming from a rich country (for only US residents get US driving licenses) so a police officer might decide that further proceedings are pointless.