There are some issues here... Though I understand (mostly) the chart, it doesn't really have anything to do with modes. The word "mode" comes from Greek, and means, roughly "mood". That is, a feeling. The modes in music that we use were developed by Greeks and have Greek names (Ionian, Dorian, Mixolydian, etc.)
Your chart shows how a 7th chord would be spelled based on the 1st through 7th notes of a major scale, transposed to the original root. But modes and chords don't really mix. A mode (for example, Ionian - equivalent to the major scale) is seven notes. So it's not correct to just list the 1st, 3rd, 5th and 7th notes, you are leaving out the 2nd, 4th and 6th.
Remember that a mode is a type of scale. The music for which the concept of mode was developed, Greek music many thousands of years ago, had no chords. It was like Indian music today, it was all in the same key, but the notes of the scale were selected to match the mood (mode) of the piece being played. There might have been a drone note (the root), but no chords.
In modern music, modes are used more loosely - for instance, in modal jazz, people often use Dorian or Mixolydian modes. The harmonies played beneath the melodies are often built using 4th intervals, or quartal harmony. Or, people just play the root chord and suggest various harmonic movements with other chords from the same mode.
You asked about a progression - I - VI - II. I would play a major scale on this progression, emphasizing the notes that fit over the chord. In the key of C, for instance, these chords are C - Am7 - Dm7. I would just play in C major over these, emphasizing the C E G and A for the first two chords, and emphasizing D F A C for the last.
Note that this has nothing to do with modes. It's not even like mixing apples and oranges, it's like mixing grapefruits and po'boys.