The problem, IMO, is that the Latin Church has been perhaps too open to new music and dropping older forms and even texts. Carolingian-era "Gregorian" Chant seems to be a likely monastic development, and does not fit well with what's known about Late Antique West Roman liturgy. Later, they added polyphony , leading to many artistically stellar pieces. After the Renaissance, you had high works of art (e.g. Mozart's Masses) as well as Masses that imitated contemporary music forms. But at least kept the same texts, even if they weren't participatory the way Late Antique liturgical music was supposed to be.
The big problem, IMO, came when they replaced the propers with hymns - no idea when they started allowing that, but probably as a counter-reformation era action to combat the popularity of Protestant hymns. Now we have musical style and musical texts for most of Mass which are left up to local discretion, often trying to elicit whatever the local musicians consider a "pious" feeling (I think this is the source of all the weirdness in Latin Catholic music).
"Why Catholics Can't Sing", referenced above, mentioned music wars in pre-WWII American Latin Catholicism, between Irish-American "folk songs" of the prevailing Irish American Catholic culture (O Lord I am not Worthy, On This Day, O Beautiful Mother) and the songs of Pietro Yon (organist at St Patricks, prolific writer, popular in pre-Vatican II USA but now largely forgotten) versus other traditions imported directly from continental Europe. The author states a televised Solemn Pontifical Mass for President Kennedy, employing Mozart's Requiem, was a cultural shock for American Catholics.
However, one of Pope Pius X's liturgical reforms was the promulgation of Gregorian Chant in the Solesmes form (Gregorian Chant had always been the official music of the Latin Church, but it was allegedly ignored for centuries and it was only Papal backing of the Solesmes restoration that brought it back into common use). Many people resisted this, but there were many who fought for it. For instance, the Society of Catholic Church Musicians fell behind it and stated, I believe in the 1930s, that many existing commonly used songs (to include Schubert's Ave Maria) were not appropriate for Catholic worship. I think the Vatican's position has been consistent - Vatican II is clear that Gregorian Chant and Latin are the normative music of the Latin Church, Benedict XVI was for a restoration, and even Paul VI gave each diocese a list of Gregorian chant music, entreating the bishops that all Catholics should know at leas those tunes.
So, I think the problem - schlocky music conforming to "contemporary" musical styles, ignorant of Latin musical tradition - has bedeviled the Latin church for at least the entire last century, at least in the United States. (though the music today is arguably more schlocky). Against this trend you have some Popes sticking up for Gregorian chant and some people following them, but for the past 100 years it's seemed to be a mostly loosing battle. However, even today there are signs of hope - 20-something friends of mine who are responsible for the 9AM Mass at their local parish use a basic Latin Ordinary, some more traditional hymns, and the propers+responsorial psalm sung in English to simple Chant tones.
Finally, I would not say the "Byzantine" tradition is necessarily better. Confining myself just to Greek usage (since that's what I know), current Byzantine chant is almost certainly not the music used in the old liturgy in Constantinople, or even the same as the dominant music at the time of Turkish conquest (though at least the texts are the same with the latter). In Greece, some of the older chanting styles are kind of hard to listen to. In the US, we have organs and sometimes maudlin, sometimes ridiculous 1930s-60s era choral settings of the Liturgy.