I had a play with a friend's D500.
I prefer it to the D750 ergonomically, in the same way I prefer the D300 D700 bodies.
However, not a deal breaker, as I use both types of body, and you get used to each one.
As for the focus, as you don't use the outer focus points you won't gain a significant benefit from the wider spread of points on the D500.
I had a play with the D500 with a Nikon 70-200 VR2, and the focus was extremely quick. But it's quick on the D750 too.
I was messing around focus tracking my border collie running at full tilt, which is really fast.
The D500 with that lens didn't miss a shot.
The D750 didn't either !
As long as you use the correct focus mode, and your settings and technique are good, either camera will focus perfectly well for equestrian events IMHO.
Also, you say you don't "machine-gun" either.
If you did, the D500 would be a clear winner, not only in frame rate, but in buffer size too.(200 Raws for the D500 !)
The main difference would be the apparent "reach advantage" of the crop over full frame. It's a 1.5x field of view difference.
The full frame sensor will give you about a one stop advantage when it comes to noise at the same ISOs.
Also around threequarters of a stop better dynamic range with a full frame sensor.
Personally, I would go for better glass, and see if that works better for you.
Just my 2p worth !