Which Gymnopédie? I'm familiar with several harmonic analyses. I'm sure they are exactly what Satie was thinking when he composed these pieces.
Why did Satie use a 2-sharp key signature?
Not the 2nd one, or were you up for a challenge?
How much were you paying them? That affects their attitude exponentially.
And your observations of Miles' transcription?
Nah. It's hardly important.
I've done all 3, and all 6 gnossiennes, and plenty of other stuff besides, but that's hardly important.
I wasn't paying them, I was working with the director in preparation for a performance, and I spotted an opportunity for something 'extra-curricular'.
As for the Davis transcription, you will have picked up from my posts that Jazz in general really doesn't interest me enough to go looking. I'll leave that discussion to you and @IamGroot .