Railroad companies having cab cars in Europe do sometimes run them in the middle of trains. It is not impossible, it just voids the benefits of having a cab car for the particular train and schedule. Sometimes it is acceptable in the greater scheme of scheduling and planning.

One example that comes to my mind is prisoner transports. I have seen a prisoner car (which cannot be passed through by passengers or crew, it has no doors at the ends) in front of cab car. In that sense, the cab car is being used as a normal passenger car, of course in the rear end of the train.

So, if Amtrak values private cars and privacy in such cars enough I think they can find a solution without people passing through. If the schedules require faster turnarounds, one low-tech solution is not selling tickets to the cab car.