This is the second time that I have lurked this thread. I still don't know what the complications are???
The organ doesn't suggest F-Ab-Bb to my ear.....this is what is being played by the organ (during his rhythm behind Cropper) and the bass. AS Tim Bowen notes....follow the bass. Cropper at times tic-tac'd the bass line...followed it verbatim in fact. When the organ takes the rhythm behind Cropper's lead on the 1964 original, the organ clearly is playing the F without the benefit of the 3rd as noted above. Then, to my ear, he is playing Ab (on guitar...1st fret on the 2nd, 3rd, and 4th strings) and Bb (3rd fret on the same strings) and indeed catching that Ab infront of the 1 of the next bar to get that Fm thing/feel. When looked at this way....following the bass....things get simpler. YOu don't have to think of that Ab or Bb as some chord over the tonal center. IT is simplly a chord change...simple and brilliant in that simplicity. IF you get the intervals arranged, the sound is there.
As for the intro chords that Cropper hits in the beginning...the stabs....I hear F, Bb and C. ...no thirds. thus leaving the minor/major thing unresolved and allowing the bass to establish the chord changes. I don't hear what is in essence an F7 for the second chord as noted by JOe K above. I hear Bb and then C.
PErhaps the confusion of the OP is in thinking that the F is the only 'chord' for those 4 measures??? But it is not....
F-Ab-Bb for 4
Bb-Db-Eb for 2
F-Ab-Bb for 2
C-Eb-F for 1
Bb-Db-Eb for 1
F-Ab-Bb for 2 and go again.
To my ear, the secret is in the formation of the second two chords in each phrase. They are full chords, and the second chord happens to be 3 of the notes that are found in the minor of the first chord of each phrase. IF one plays those chords based on the barre E form, the sound isn't there. IF one uses the notes that I showed above, then the sound is there. AS for 'playing a Bb triad over the F', I have to disagree....and the bass line disagrees, also. It is a Bb triad played over a Bb, and an Ab triad played over an Ab.
The F is not a whole note, in other words.