Mobility even in areas where public transportation cannot make profits is an important part of quality of life, and that should not imply being forced to own a car. But does it have to be Amtrak? Can't the long distance services too be put out to tender, with the best bidder getting the contract? I'm sure if the government paid their losses, a lot of railways would gladly take up passenger service.
However there is the problem of infrastructure. You cannot offer good competitive passenger rail on tracks already heavily used by freight trains, typically at lower speeds. It would require building new high-speed, high-performance lines, and that of course is a task for society, as is the building of highways. To invest efficiently, first put up a target timetable, seeing which speeds will be necessary for a good synchronised timetable and what number of trains per day/hour you want to operate. Then of course, don't just build the tracks, but actually run the trains, and do so over a longer period of time, allowing people to get used to the mode of transport. (Too many ambitions train systems have been cut down after a few years because they didn't initially catch on ....)
Trains are also vastly more energy efficient than short distance aeroplanes, and better than cars as well, about the same as buses. As energy prices go up, the other modes of transport will become more expensive faster than trains do, where a lot of the fare goes into manpower, not into energy cost.