"For instance, you can always chop up notes into chord tones vs non-chord tones; and then separate non-chord tones into passing tones and neighbor tones. These are very much just classifications."
This is not a great example to start with, as it presumes that chord tones had meaning in the first place. A better example would be "why are chords generated from the pattern (1, 3, 5) to start". By they way, what's wrong with classifying things? The class division has meaning. Are you opposed to major versus minor?
There is a tremendous amount of information to unravel to get to an answer. Any fast and loose claims that it is or is not scientific are probably quite thoughtless.
There is great significance in the structure of the diatonic scale and the significance of the I, IV, and V chords within that scale. They are in fact related to the natural harmonics in a harmonic sequence. This is the type of sequence you get for a vibrating string or standing waves in an air column. In contrast the natural harmonics of plates and other vibrating system do NOT follow this sequence. Many musicians go through life thinking that all vibrating system obey f_n = n*f_1 but that is not the case. There are many exotic "overtone" sequences. So why then does most of Western music seem to gravitate towards the "harmonic sequence"? It could have something to do with biology and how we are constructed (by evolution). Our ears, and brain, are naturally tuned to this series of tones. The nerves in the ear will self excite in the harmonic sequence when a pure tone is introduced, this is called aural harmonics. The brain will think it hears a "fundamental" that is NOT present in the acoustic field that excites the ear due to a process called fundamental tracking. Both of these phenomenon seem to suggest that nature has favored the harmonic sequence above all others. Why? I really do not know but would guess that it has to do with our use of vocal calls to communicate and the fact that our throats (or throat + head + vocal_chord system) are approximately described by tubes, which obey this sequence.
If we had evolved differently we might actually detest harmonics and the natural sequence of tones produced by them, e.g. the (1, 3, 5) which is the foundation of the major triad and the diatonic scale.
Much of what the general population finds "pleasing" about music (or in contrast "unpleasant") is related to combinations of tones that have supporting harmonic structure. The German physicist Herman Helmholtz set out to provide a physics based explanation for the concepts of consonance and dissonance, as well as devices like authentic and plagal cadences. His thesis was that these things are not merely one of many choices but essentially mandated by physics. You can read about it in the text "On the Sensations of Tone" which is still in print today. This is the foundation of musical acoustics and psycho acoustics.
Based on his work it might seem that the rules of music theory are in fact grounded in real science and are the only rules any human would arrive at given enough time and opportunity to explore all possible avenues. However, I do not think this is true. For one thing Helmholtz started with the premise that Western music theory was in some sense superior to that of other cultures, if it even existed in other cultures. In other cultures, Indian music comes to mind, different scales are used that do not necessarily have tones related to the harmonic sequence. They also lack the use of multi voice harmony, which is really where the rubber meets the road with respect to true "theory", e.g. the rules of harmony. Harmony is what creates the opportunity to hear dissonance and the need to resolve it to consonance with a resolution.
I know this is not really what you are asking about but I do not think it is possible to completely separate these concepts. We need to correctly identify them to understand how they are related but we cannot divorce them from one another completely.
The specific ideal of chord tone versus non-chord tone is significant as much of music has developed over the centuries or millennia so that we tend to favor harmonizing melodies in such a way that the notes are a chord tone. That is rather than letting the melody move over a drone, if the notes leave the set of notes in one chord we move along with it by changing chords to support the melody. This type of process creates the feeling of movement. That does not mean that musicians MUST move with the melody by changing chords. The composer that holds a chord while the melody moves in and out of the chord tones is choosing to say something. They are creating a tonal landscape that may produce a feeling of conflict (that is not negative). Whereas the composer who move chords with the melody is creating a feeling of support. The fact the these two different approaches create different feelings in most listeners indicates the importance of having a language that helps us distinguish them.
The rules of Western music theory have evolved over time to reflect what might be considered best practices, those ideas that seem to work more often than not. That doesn't make a perfect system, and it needs to evolve as we learn and create more. But the fact is that it is a good starting point for developing musical ideas. Artists that go to university will learn about color, light, shadow, perspective, etc. And these are all very scientific. But they don't tell the painter how to paint. They merely provide the painter with a rubric for developing a visual piece that would indicate to the observer that there is a light source somewhere off in the distance, not seen in the painting. The painter is not saying "the rules of light and color tell me what and how to paint" but they are saying "if I want to convince a person that the following things are true here is the trick I need to play". That is how theory is used in the arts. If a painter wants to destroy the idea of perspective to make a political statement then they can choose not to use the mathematically precise techniques that create realistic perspective. Music theory works in much the same way. The rules as we know them have some grounding in physics but as artists we are fee to break the rules at any time. I might want to make the listener feel a sense of emptiness at the end of a song. To do this I might choose NOT to end on I, or I might end on a suspension. Music theory doesn't say "you must always end with a V7-->I with the I voiced in root position and Do in the melody" it does say "The aforementioned device creates the strongest sense of resolution from among all possible chord progressions". That last statement is on pretty firm ground (but perhaps still only in terms of a general consensus).
"Thus, it seems that, to a large extent, music theory is only giving us a language to describe music. It doesn't seem to have significant "scientific content" in the sense that "music theory predicts what is good music". That is, it isn't true that "only music written according to XXX theory sounds good to an untrained ear". I emphasize "untrained" because people can be trained to prefer one style of music over another."
So, first of all "... a language to describe..." is exactly what any theory does. I think you might be misusing these terms. The foundation of music theory is vocabulary. But that vocabulary is significant. Even before we had a false theory of gravity, replaced by another which is probably also false, we had a rubric for explaining planetary motion based on observation. Most of science is in fact "stamp collecting". We observe, take notes, and develop recipes for recreating things that work. Just because someone can come along and be contrary, stating "well I don't like what everyone else likes" doesn't mean that the rubric is useless.
Second, your use of "trained" versus "untrained" is deceptive. The whole point of music theory is to provide a set of practices for producing music that would appeal to widest audience. The greatest works of music throughout history were written for the masses (or masses, i.e. masses attending masses). These "rules" are not based on what one or two musicians thought was best but what gets the most people in the door to pay for your music.