This is very niche, I think. People with private cars used those not so much because they couldn't afford an engine, but by the fact that they had no real access to the railways in general. In contrast, we have private boats, private planes, etc. Fully capable vessels, rather than a carried vessel.
Hmm, can't speak to European practice, but in the US the economics and motivation were a bit different.
First, they were high tech comfort wagons, often having amenities years if not decades before general travel did, and often brought along their own servants and chefs.
Secondly, it allowed them to maintain distance from the hoi polloi in coach and Pullman class service.
Third, they had much more generous individual space and comfort for their bedrooms, drawing rooms, etc. that commercial cars would not have.
Fourth, it was a status symbol that was much more practical then a yacht as to speed and possible destinations. The ultra rich often had both.
But fifth, and perhaps most important yet likely surprising, they were common among the rail set because they rode free.
Most private car travel was by rail moguls who got courtesy hauling of his railroad's business car by a fellow rail baron's road, and the favor was returned. Or just as likely, the car was owned by an important customer/potential customer and so got a courtesy trip as a sales tool.
The end of private railcars occurred when the same sort of railing against privilege such as the three martini lunch, made a rule that all private cars being hauled had to buy 18 Pullman tickets to pay for their trip, no gratis.
The cars were laid up and largely gone in a generation, decades before Amtrak.
