There are clubs[1][2] of owners, and they'll generally rent them out to people. We looked into doing it for my bachelor party. Unfortunately, the cost is akin to renting a yacht for the same amount of time (On the order of thousands per day, minimum), so we quickly shelved that plan for an AirBnB.
I don't have personal experience, but I've heard it's not viable. The biggest issue is that Amtrak offers the service on a "best effort" basis, which means that if the train you want to hook up to is running late (which this frequently are due to conflicts with cargo traffic), they won't hook your car up, and you have to wait for the next train, which also might not be able to hook you up.
Part of doing this is knowing the rules and the “real rules” - you have the car attached at an end node (Los Angeles or San Diego, say) even if your group gets on in Burbank.
There are a few clubs that have cars that do this for a club outing. Members pay a small amount of dues, but the largest cost is labor - you are expected to help rebuild their cars. Most of the club money seems to come from renting the cars out.
The above is what I gather from reading their websites. However there is no club close enough to me for joining to be reasonable and so I didn't verify the above.
Somewhere, the locomotove nomads travel the wastes of North America in their reinforced rail cars. They never speak to one another, but sometimes you can glimpse the deisel smoke of a distant train on the horizon at sunset…
[1] https://www.aaprco.com/
[2] https://www.rpca.com/