I absolutely love your last paragraph there, Mark!
A similar analogy to playing the piano is the fact you're touching upon; if, say, you're working on a virtuoso piano piece and insist on playing it fast all the time, you're actually getting better at making the mistakes that occur.
When I taught piano at the local conservatory for the last decade or so, I developed a kind of pyramid-shaped study pattern for my students. A wide base of slow practicing, a middle part of "not too slow / not to fast", and then a small amount of top speed (often exceeding the metronome markings) at the top of the pyramid.
It worked amazingly well. If you only practice slow, you'll never get the economy of motion that's needed for fast playing. So to me, the best way is to do it all at once, but in the proportions i mentioned. Fact is, if you do the really fast practicing a good amount faster that required for the piece, the students just laugh at first, but because it's so ridiculously fast, they actually often execute it sooner than they would if they practiced at the standard tempo for the piece! All this because they drop their anxiety of playing perfectly, and this has been a real eyeopener for them.
I think much of this pyramid scheme (pun intended) could be carried over to tuning technique as well.