Where did you ever get the idea? …nonsense!
I read that as room in the budget. By purchasing a D500, he would possibly have some money left for lenses whereas the D850 will cost significantly more up front.
To the original poster - I own a D500 and a D810. One of the differences between the two is the frame rate. The D500 can capture more images per second. However, I find that I rarely use the higher speed. It just produces too many "similars". I can generally capture what I want at a slower frame rate. The primary reason I take the D500 out these days is for the additional "reach".
For the situation you've described, I don't think you would be disappointed with the D500. Invest the difference into a good 70-200mm f/2.8 lens. It will work equally well when you later upgrade to a full frame sensor camera.
If you really want full frame now but don't like the price of a new D850, consider a used D810. They are readily available lately in the under $1500 range. The D810 is reported to be less capable at focus tracking than the D500, but in my experience your technique and amount of time spent practicing will have a much bigger impact on your keeper rate than the camera technology. I get about the same percentage of keepers at the race track with either camera.