> Use the correct fingering, dynamics, articulation, etc. from the very beginning; you are never going to get around to “fixing” it later.
When I was an impatient kid my piano teachers would keep insisting I play sections slow on a first pass, which annoyed me (it was like reading a book out loud slowly), but it was only until many years later when I played pieces like Un Sospiro[0] which are intractable if you do not start practicing at something like 16 times slower than performance tempo. If you start out correctly, the rest will follow, and IME, sight reading greatly improves as well.
> [...] you may reach a point where you can’t play any faster, no matter how hard you try. What is likely happening is that the hand motions you are using—which come naturally at the slower tempo you started with—simply will not work at such high speeds.
This is completely true, for instance, in ragtime, the left hand makes big, quick jumps and there is often no time to look at where it will land (which one likely does when playing slowly), so repeating a bar over and over again to incrementally refine your distance estimate is crucial.
This applies to things like typing on a keyboard as well, if you focus on correct finger placement (and correct use of the 4th and 5th fingers), your WPM might take a hit for a few days or a week, but in the long term it really helps.