Correct, privately owned except the North East Corridor. Not all the freight railroads are created equal when it comes to dispatching Amtrak.
I'm thinking of a ride on the California Zephyr (Chicago to Sacramento return) in February to enjoy travelling the Rockies and Sierra Nevada's in winter which doesn't have a great on time record.
Here are some stats and which railroads cause the most delays.
Delays can happen for a variety of reasons, but the leading cause of delay to Amtrak trains is “freight train interference.”
www.amtrak.com