It appears this tune is in A Major.
I'm only guessing, but I think this is why you have trouble understanding what's happening. Perhaps you assume that because it's declared in the key signature, the entire song must be in a single key from beginning to end, and so you should be able to apply the same explanation patterns of simple functional harmony throughout the song?
In which key would you say this tune is "in"?
(I don't know where that came from but maybe it's a secret recording from a Pat Metheny Group cover band's rehearsal, an unpublished song maybe!? I think they should just keep playing covers if their original tunes are like that.)
What I'm saying is, particularly in jazz, you should treat each moment as potentially being in a different key. Jazz harmony often plays around with the feeling of changing tonal center and mode. Often the key signature does not reflect the entire song from beginning to end, and it may be a compromise that's chosen just to get the notation done, not to make any hard claims of the overall key. Otherwise they would have to write a key signature change or two for every bar. It would look awkward, and maybe it can be assumed that jazz players can tell where their home base is at each moment even without giving official recommendations all the time. :)
So, play the harmony and toy around with it. If you press "pause" at any given point, where would home be at that instant? How would you solo over it?