Mizmor - I personally don't ascribe to the "specific chord symbol always implies a specific scale or mode" approach to jazz harmony, so unless I'm playing a modal tune that was written with a particular mode in mind, I'm not usually thinking about modes when building walking lines over standards. I also don't like to think of most scales as having roots that you have to recalculate to a new pitch level every time a chord changes. Instead, I tend to hear in terms of key centers for larger combinations of harmonies in most progressions.
It's a subtle distinction - and may be one that could be confusing for a beginner - but I think it's an important one. If you take the chords of the first 8 bars of rhythm changes literally as producing scales/modes with no idea of the context of the key of those 8 bars, you might come up with something like:
Bb Ionian
G Mixolydian
C Dorian
F Mixolydian
D Dorian
G Mixolydian
C Dorian
F Mixolydian
F Dorian
Bb Mixolydian
Eb Ionian
E Whole-Half diminished
Bb Ionian
G Mixolydian
C Dorian
F Mixolydian
Even if you are playing the tune relatively slowly at say, Q=120, thinking this way means you would be thinking of a different mode every single second you are playing. And at Q=240, you would have to be thinking two modes per second for the duration of every A section of the song. IMO, this kind of paradigm is not the way to begin learning to play rhythm changes.
Instead, I tend to think of each chord as containing 4 notes with a possible stepwise note in between. The notes in between are ear driven and therefore subjective, but if you look at the progression as a whole, you can see that the whole 8 bars is really a big turnaround in Bb Major that modulates briefly to Eb in bars 5-6 before reverting back to a Bb Major turnaround. If you count only the notes contained in the chords, everything comes from the key of Bb except a few notes which lead to notes in the key of Bb (i.e. - the B natural in the G7 chord leads to C, the Ab in the Bb7 chord leads down to G; in the E07 chord, the E natural leads up to F, and the Db can resolve either down to C or up to D).
This may seem even more confusing than the modes, but to me it's easier to hear one big tonality (Bb) with toggle switches raising or lowering certain notes of the tonality for the duration of certain chords than to think of one or two different modes - each with a different root - per second.
When in doubt as to the active key center at any given time, the melody is usually my first and biggest clue. In the case of "I Got Rhythm", the original melody in the A sections only contained the notes Bb-C-D-Eb-F-G-Bb.
Hope that wasn't too confusing!