Classical is the mother of western music ... it has centuries of historical value. And in Classical, there is actually no such thing as actual chords. In classical, a combination of notes is harmony ... and it can be quite complicated.
Chords is also called "Mordern Harmony" ... which in a way is a short cut to classical harmony .... it's also simplified and more flexible to suit the needs of the music of our day.
Regarding the examples you gave ... you are right to say following a C chord, we may play C Ionian, E Phrygian, or A Aeolian. But in a real situation, i dun think there is time for so much evaluation .... because E phrygian and A Aeolian is the same as C ionian ... there is no point in complicating things and confuse yourself with more names.
Instead ... you should simplify things by referencing everything to the respective major scale. Modes are actually meant to suit the function of the chords.
example 1: A C maj chord can function as a 1, 4 or 5 position, Hence Ionian, Lydian and Mixolydian.
example 2: A C Min Chord can function as 2, 3 or 6 position. Hence Dorian, Phrygian or Aeolian.
But when playing these modes ... instead of thinking aboout modes, it's relly easier to reference it to it's respective major scale because in actual situations, you have no time to think about modes .It's faster to think abouot Major scale ... but you know which mode you are playing.
example 3: When you play Cmin7, F7, Bbmaj9 .... you are playing C dorian, F Mixolydian and Bb Ionian. But you are actually playing Bb Ionian .... is there a point to think about Dorian or Mixolydian?
example 4: |E maj7 Ebm7| Db mm7 Bmaj9| What major scale is that?
example 5: |Am7 G#7(#5)| Gm7 C9| Fmaj7|Fmin7 Bb7|Eb6/9| This is a combination ... change of key. Hence there is more than one major scales involved. Can figure out what that is?
And for your analysis about Jazz progression, jazz solos are more than just scales. Scale analysis is only one such tool to define a jazz melody. But Jazz melodies are much more intricate with combination of scales, chord tones, passign tones and tensions. And you can never define a jazz melody under a scale becaue jazz melodies are much more fluid and unconstrained. A bit too much to cover within this forum. If you are interested, i can tell you more but for now, my fingers a bit tired ... hehe ... so have fun with the chord analysis.