There appears to be a confusion about harmonization of melodies. Melodies that are definable in so-called "Western" or "Occidental" music for the most part contain harmonic ideas. The fact that some haven't mastered the harmonic structure enough to effectively
highlight these ideas doesn't mean that the melodies are essentially injured by harmony.
All this means is that some who are trying to harmonize these melodies don't have the
musical background to do them justice. Most of the "Europeanized" music lends itself well to harmonization with the above caveat.

Asian, African and Native American music would tend not to be harmonized since it is not
in the tradition of the culture to do so. Harmony is not an aspect of this music. One reason is that quarter-tones are employed which are not "harmonizable" in any sense
of the word that we know of with a Euro-centric view of music.

Even the "so-called" "Church modes" contain harmonic information although not as advanced as when Bach came on the scene. There were many chords that were considered "dissonant" for the early period such as a dominant-seventh chord. Now,
the harmonic pallette is so varied that many chords that were considered dissonant in the early days of music are now commonplace.