So, they don't consider weight?

If your private car is "that" heavy, and they need to hook up an extra locomotive just for you, then that is going to be a steep fee per mile added.

They actually mention it in the PDF under "more locomotives" or something.

I think that mainly applies when you add a whole train to the train (you can hire them for multiple private cars)

It's a passenger car, it weights basically nothing compared to a coal car so they probably just assume "some default" weight.