The TAS does a few special (virtually impossible for humans) tricks that render the finish line warp useless. The reason he doesn't do it on lap 1 (where he does the regular shortcut that is more commonly seen) is because he needs to get to the items in order to get that blue shell (through a finish line glitch and a boo) to do the shell skip at the start of lap 2. Then he skips the jump altogether as he CPU jumps to the road, and then does a special skip that abuses checkpoints to skip a lap.
in terms of human runs, the finish line warp is best used on lap 3 since it ends the race once it puts you down. On laps 1 and 2, it's hard to take advantage of because of the position lakitu sets you down (you'd need to quickly spin around and then accelerate) so it saves very little time if any at all. As Hubz0r said, it's a hard skip to hit, therefore you don't see it in too many runs (although some people do go for it as failing it typically still results in getting the normal skip anyways).