I agree with all the criticisms of Amtrak that I'm hearing in this thread and that are common for Americans. But I want to tone down the love of European trains a bit. Amtrak is much much cheaper than trains in Europe and in general you get more legroom. European trains have shitty food, as does Amtrak. Actually the best food I had on a train was in the early 1990's on a trip from Warsaw to Prague on a soviet era train. It was actually cooked in a fancy dining car.
The UK has wrecked their train network. Delays are terrible and the network doesn't receive any real new investment. I blame Thatcher. A few weeks ago there were no trains between London and Leeds over a bank holiday weekend. For 3 days there were no trains between London and Leeds. This affected me and I had to change my plans. I suspect that this affect all travel north out of London for 3 days. How can you rely on a network like this? Sure they give you advance notices, but where is the redundancy in the network?
Dutch trains are better, but they're mostly just for commuting. Distances are short so the Dutch rail network is basically what Americans would consider a suburban light rail network. Like privatization everywhere else, the private elements of the Dutch rail network play games with timings and commitments to 'juke the stats'. For example, a late train may be cancelled because the penalties for cancelling trains are less than for a train being late. Yes, this actually happens.
I have less experience with German, French, and Italian trains. But at different times in my life I have been screwed over by them as well. Delays, cancellations, confusing rules that only the locals understand. Trains in these countries are still considered a second rate service compared to airline travel. And how exactly am I supposed to take a train through Paris exactly? Let's say I am in Belgium and I want to go to Lyon. How can I do that without taking the Paris metro or taking some stupid long route? Answer, you can't. Because trains don't go through Paris.
So in short I don't want Americans to think that European trains are all that great. They're much more expensive and come with their own problems. They lack redundancy and delays in one part of the network can cause problems in other parts of the network.