After the Renaissance, Western music shifted from a horizontal, linear approach (the beauty and integrity of each voice is important) to a more vertical approach (each note of a line is interpreted for its relationship to concurrent notes of other voices). Gregorian chant, which the basis for Western sacred music prior to this moment, freely utilized all 7 diatonic modes, because it relied only on the shape of the line, without having to consider any vertical implications.

When music is considered vertically, any note is definable by its harmonic function. The functions become codified, which partly explains the reliance on Ionian and Aeolian modes. These two offer simple and clear ways of resolving the tritone between the 4th and 7th degrees of the diatonic scale. The tritone became associated with "tension", and its resolution could be more easily handled by treating Ionian or Aeolian as the tonic.

Within modes such as Locrian and Phrygian, the 4-7 tritone seems to reside comfortably in its "unresolved" state. It would seem that after and during the Renaissance, taste and fashion shifted away from finding this "unresolved" quality enjoyable.

One more idea. Let's accept for a moment that the most characteristic scale degrees of any mode are the 1st, 3rd, and 5th. Then let's assume we wanted to isolate the effect of the tritone that exists between the 4th and 7th degrees (in Ionian mode in C, the F and B). Then, the only two modes that DON'T use the either the F or B in the 1st, 3rd, or 5th position are (you guessed it) Ionian and Aeolian. These two modes allow the tritone to serve as dissonance (lack of resolution), which gives the primary tones the quality of feeling resolved.
This is a fact based on my own observation, not research.