I have a very specific question that is kind of related to another question. What countries in the Schengen area accept US travel document (form I-571)
Assume that someone has a Form I-571 Travel Document issued by the US and no other country's passport. This link looks official and seems to indicate that Germany doesn't require a visa for such travel document holders to enter the country. The same for Netherlands, even though the Visa Advisor tool was not working when I tried to access it.
And, assume that they are mainly visiting Germany with a day or two short trips to the Netherlands and Belgium by train from Germany. (Belgium and the Netherlands allow visa-free short visits for such travel document holders according to many sources.)
My question is, if that someone enters the Schengen Area (excluding non-EU territories) from the US via Germany by air and wants to take the train to Paris to Brussels and then to Amsterdam and finally back to Germany, would there be any legal basis that allows or prevents this trip? The only unclear bit of this trip itinerary is the train ride between a train station in Germany and Paris and then Paris to Brussels. Please provide as much credible sources as possible about the legal grounds for your answers and explain the risks. Information online has been mixed/inconclusive.