One of my favorite quotes from my first NGA teacher: "Slow is smooth. Smooth is fast." It's a reference to exactly what you're talking about - using slow movement to get it right (and smooth) so "normal" speed is faster.
I first heard that in the military. It goes hand-in-hand with the "crawl-walk-run" method we used as a way of learning, practicing and mastering skills. As a competitive shooter, I have been making use of it as well, focusing on slowing down, relaxing and getting shots on target. Interestingly, most of the better shooters I know use a combination. They use the "slow is smooth, smooth is fast" mantra, but they will also shoot events other than the big ones at the ragged edge of speed. The theory being they push it to the point of missing, making mistakes in practice, then slow it down just slightly for the big matches. It seems to work.
In regard to kata (and techniques), since returning I still feel I am in the "crawl / walk" or "slow" phase, trying to smooth things out so I can get to up and running. This thread has served as a reminder that I do need to focus on doing both those things.