I drove the SE-R at the Newport ProSolo last weekend (looooong, faaaaaast, narrow), and it had a pivot cone. The fastest way around it was to go to first and take a tight exit, even though you had plenty of space on the return route to place the car. I worked that section, and a lot of people were going wide and holding second in the early heats, but tight and downing to first was the norm later. Especially among people who made it to the Challenge round. Sometimes less distance and lower speeds in really tight stuff is better than carrying more speed and driving a wider entry and exit, but we don't see this often, except for turnarounds.
An autocross run entirely in first gear sounds like pain. First is so low that you're just bucking and spinning wheels the whole time. At least we have limited slip of some sort.
On second gear courses, leaving it in second is also nice because you can left foot brake without having to dance on the pedals. As soon as I shift out of first, my left foot is over the brake pedal.
I don't have my Watts book at my desk to refer to the naming, but for steering I taught myself to shuffle steer ages ago. Hand-over-hand is clumsy for something as busy as autocrossing. With shuffle steering, pulling down on the wheel to turn instead of pushing up is more precise and less fatiguing because it uses larger back muscles more instead of finer arm muscles. It's also elegant, and impresses passengers. Practice that for a while, and you can become a natural.