No cite handy, but I believe that since its formation around 1971, Amtak, the national intercity carrier, has only had about 1-2% of the intercity travel market. Commuter operations, most of which are managed by local metro transit authorities, have a higher share.
The reasons for the low numbers are legion, but mostly derive from the fact that until Amtrak was formed, rail passenger carriage was the responsibility of private railroads, rather than of a nationalized system, and theoretically run as a for-profit operation. Thing is, when people began abandoning rail for air (faster) or automobile (more conveniient) travel in the '40s and '50s, most railroads lost interest in money-losing passenger traffic. Passenger trains began to be seen by them as impediments to getting freight over the road, and most railroads began to eliminate passengers trains as fast as they could get permission from the government.
Amtrak came about mostly because if it hadn’t, there likely would be no, or very little, national long-distance passenger rail service. For all its life it has been a) dependent on government subsidy for its a large proportion of its capital; b) a political football with its year-to-year funding subject to the whims of Congress, and c) dependant on the freight carriers for access to most of its right of way outside of the Boston-New York-Washington corridor, which it owns.
Today, Amtrak operates a ridiculously sketchy national service, simply because that’s all it can afford, even with subsidy. Train frequencies are simply not enough to make it a practical travel option in many areas (Houston, with one of the largest metro populations in the country is served by one, train, three days a week!) but the government would likely take a lot of flack if it were to increase subsidies enough to expand service to a useful level. Meanwhile, while the freight railroads are obligated to give Amtrak trains priority by mandate, they often fail to do so; many trains, especially in the west, are routinely hours late more than half the time.
Although the carrier has managed to stagger along for more than thirty years, it has come close to shutting down a couple of times since 2000. Without a steady source of capital, and a commitment to on-time service, its position is prbably untenable, and one is likely to see major changes to the system in the next few years: most likely a shutdown of the national system, with certain routes turned over to state governments to operate if they wish. The Northeast corrider is the most economically viable part of the system and would likely remain Amtrak-operated, were the above to happen.