Your problem isn't the shutter speed. 600 froze the wheels so it would for sure freeze the rider. The problem you are facing is the slow AF of a lower end lens. I don't know much about Nikon lenses, but if they are comparable to Canon then the 75-300 is a value lens and is slow to AF and probably a bit soft at the 300 end unless you get up around f8 to f16. I used to have the Canon 75-300 and the pics looked nice, just didn't have that killer cut yourself sharpness that I wanted. I sold and got the 70-200 f2.8 and have been very happy with it, now I just need a camera that can AF track as fast as it can. (have a 20D, want a 1d mark 2). That leads me to another thing to think about. A person or rider coming directly at you is very hard for the camera to keep up with. I deal with this a lot shooting basketball from the baseline. I don't know where your camera ranks in the Nikon lineup, but the camera's AF just may not be able to keep up. Most likely the lens is your initial limiting factor though.
For information a great place to go is www.fredmiranda.com
and visit the forums. Also, www.wheelsandwax.com
is an extreme sports photo site.
Rich.. how fast is your friend going on his cart that you will have to use 1/600 to pan? I was panning cars going 70 on the highway and I was down around 1/25 to 1/50 and getting pretty nice blurs. My first shot was at about 1/100 and I got very little blur in the ground.
Brad, I hope that helps you a little bit. Fred Miranda also has gear reviews so you can read up on what other people say about the 80-200 before dropping the cash.