A retired transportation engineer I know often talks about how frustrating it was to deal with "efficiency experts" who wanted to build, staff, and plan the railroad/terminal/whatever for the ideal case. They'd set up a system that would move a little more freight a little faster for a little less money when it worked, and then totally screw up the railroad/terminal/whatever for hours/days when it failed -- or at least it would deliver the lucky carloads a day faster, but regularly deliver the unlucky ones a day or two slower. All because the slack -- the resilience -- wrung out of the system by people who never really looked at a train/ship/terminal/piggpacker/etc. (I could make an analogy to current hospital capacity policy, but that might get political.
Just-in-time does sound like a good description there. Since just-in-time manufacturing/distribution/etc. is all about reducing inventory to the barest possible minimum, it depends on scheduled deliveries that are predictable over the relatively long term and/or on the manufacturer/distributor being able to get unscheduled shipments in and out on short notice. Neither of those sounds totally compatible with PSR, even though they come out of a similar notion of how to maximize profits.
Back to Amtrak and PSR:
In the age of smartphones, online ticketing, few or no paper timetables, why not set up Amtrak LD schedules like this: Fixed departures from origin and from one or two big intermediates; fixed arrival at the endpoint. In between, the host railroad can have the train run early in some places, late in others, according to its needs. When passengers book from intermediate stations, they get a three-hour window (e.g. 12 noon +/- 1h30m). A couple of days before a passenger is due to board, they get an email/text/call telling them a half-hour window, based on the railroads PSR plans.
The railroad wins because it can treat the Amtrak train more like a unit train, allowing it to gain time on a part of its run where it's possible that day, and use that to offset a known or possible delay further along.
Amtrak wins because it gets better OTP to endpoints and major intermediates, and because there are not so many passengers complaining about late arrivals as a kind of broken promise.
Passengers to/from major intermediates or endpoint win because they get better OTP. Passengers to/from little intermediates win because they (or their rides) are less likely to show up at the station two or three hours before their train.
Or maybe I am totally mistaken. I talk so much that that happens a lot.