I've used my automatic on numerous tracks including Sebring, Daytona, Roebling Road and others. I chose it because I'm too linear to be able to learn the "line" while heel-and-toe and shifting (my brother in law told me once that I'm so linear I could easily win a can't-walk-and-chew-gum contest). So I have a GO pedal, a STOP pedal and a steering wheel to manage and can learn the line much more quickly than if I had to add in the other activities.
It did take four transmissions to learn about preparing an automatic correctly for road race use: maximizing
oil flow in the converter and transmission, proper cooling and finding builders who understood what I needed. After that, I have a lot of hours of smiles and more than 30K of road miles behind the car without further problems.
In the right circumstances I've been able to run with newer 6-speed Corvettes and similar cars just fine. Oh, and the brakes need to be very much equal to the task too!
Jim Hall started this kind of thing way back and it seems to be a good way to go. If you choose to do that, enjoy!
Tom