As far as I'm aware, Amtrak can delay a train indefinitely - the host RR likely just wouldn't be penalized for delays as the train entered their system late. Mechanical issues can and do happen, and I'm not aware of any instance where Amtrak was refused passage on a host railroad due to the delay, even at an origin station. Many trains also pass over multiple host railroads, and while I don't know the specifics of the agreements, Amtrak can certainly enter onto another host railroad's territory even when the train is late due to delays on the previous host railroads.
The main reasons Amtrak had started to take a position to not hold trains in Chicago was because they wanted to reduce delays overall (ideally if the train is on time leaving the origin station, the host railroads will get it across at the scheduled times.) There could also be concerns about missed connections downline, although now with the three-a-week schedule I'd hope that Amtrak would be more concerned about getting passengers through Chicago (and not having passengers stuck in Chicago for two to three nights) than missing connections in other places (which generally have daily or better service, so a missed connection there isn't as bad.)