Can you imagine a TGV or ICE stopping to hook up a private car?
I've seen multiple operators (some of them entirely private, some entirely state-owned corporations, some combinations) combine their trains on an EuroCity-labelled line more than a few times.
I have a friend who is working on founding his own train company and he's actually providing services to the state-owned corporations where they have him operate their wagons in his combined train, with his own locomotive. He operated a major intercity line several times this way (they use his service when there's a problem on the regular route) - and it stopped to recombine several times. I've personally been on a train ride that combined his private for-fun party wagon with an EC train.
Luckily Europe has high speed rail which does not allow your friend's party wagon to disrupt the schedule. You might imagine a TGV or ICE picking it up, but I don't think it will ever happen.
BTW so far my friend's train was never delayed by his own fault, not sure what problem you're talking about - there's no problem, his train had usually the lowest priority (it costs money to have higher priority) so he's the one waiting for "normal" trains.
When I said there are delays, I meant the "normal" trains primarily. But sure, as he increases operation it's going to happen to him too.
Improve schedule planning and Amtrak won't have problems too.
Amtrak only owns ~630 miles of tracks out of the 21,000 it uses, and basically all of that 630 is in the North East. This is definitely Amtrak's best area, although that's partially because of the amount of people going between Boston/NYC/Philadelphia/DC; the rest of the US doesn't have as many big cities that close together.
> multi-state 1000km trip
If you get on I-5 North in Los Angeles and start driving, 1000km later you'll still be in California.