If you don't need a zoom, the AF-D version of the 300mm f4 or even f2.8 is a fantastic lens for the price.
Shirt of that, it's either the 70-200 f4 (used for a better price) or the 70-300 VR or VR2.
I had one of the first gen 70-300s and AF speed was stupid fast for a non-pro lens but wasn't super sharp from 250-300.
I got a Tamron 70-300 OS and is great throughout the entire range, but AF speed isn't much quicker than the 55-200.
In the Canon world, the STM version of the 55-250 is a big step up from the previous IS2 version and their 70-200 f4 IS is an incredible lens for the price.
The D500 and a 70-200 VR would be my ticket for race photography. The D500 AF is so much faster (with a fast lens) than the D300 or D700.
I had a first gen 80-400 VR and while the lens was great, I didn't think the AF was very impressive. I have not tried the newer version.
The other tip for race photography is to pre focus at a certain spot and switch to manual focus and work on getting your timing right. That saves ALL the AF time.
Depending on how handy you are with post processing, another trick is to shoot at 'too fast' of a shutter speed to ensure clear but dark shots, then bring up the brightness and contrast in Lightroom or Photoshop.
I have a pro in the family that used to shoot Indy and F1 and he said that's the way he always got good shots. Of course, he was always outdoors in good light so could hover around F8 and F11. Always full manual.
He had one camera with a long zoom (or prime) and pre focused and had another body with a midrange zoom set to AF.