Not correct. The trains did NOT run "perfectly well". Rather, the railroads were losing a tremendous amount of money running them and wanted to discontinue all passenger service. Just before Amtrak, most remaining passenger train service was pretty bad. Remember the Penn Central -- they operated the majority of the country's trains.
As mentioned, to save the trains Amtrak was set up as a govt agency. The infrastructure -- tracks (NEC), stations, shops, locomotives, and cars -- were old, poorly maintained, and terribly worn out. Amtrak is not perfect, but it's better than what we had in the late 1960s. (The once crack train, the 20th Century Limited, ran 12 hours late in its last years, for example).
We often read about scandals involving toll road and airport government authorities. Yet no one is questioning why the govt is running them.
As to your other question of "who is in charge", as others also pointed out, the CSX railroad (CSX doesn't stand for anything) is responsible. It was their derailment, after all, and their railroad.
It certainly is frustrating to have people stranded so long. But in isolated areas, you just can't have people abandon their luggage and climb down to the tracks. That climb down to the lower roadbed in itself would be dangerous and difficult for elderly people. Then, in this cold weather, where are they supposed to go? I think we need some more information about the particulars before drawing conclusions.
P.S. I forgot to mention that the private capital the railroads used was taxable, while the bonds used to finance roads and airports was non-taxable and often guaranteed. This made financing costs for public bodies much lower than the railroads, and gave them another advantage.
Further, the govt was very heavy-handed in railroad regulation. Railroads were forced to run empty trains for years at great losses in the name of "public service". Because of that, the railroads wanted out of the psgr business even in the few places where a psgr train might be profitable.
By the way, government regulation was partly to blame for the NYC telephone service crisis of the 1970s. The government ordered NY Telephone to hire unqualified people for craft positions and these people couldn't do the job (per Oslin's book). Accordingly to Oslin, adverse government regulation was largely to blame for Western Union's failure, such as ordered WU to assume the problems of the Postal Telegraph Company and give in to high union demands.