Firstly, it's modal harmony, not diatonic. In this case, dorian mode in A. If you've every improvised in say, a whole tone scale, you notice that you can play almost anything and it sounds good. Modal music works in a similar way. A lot of the things you find in diatonic harmony: the tension between tonic and dominant, chord progressions in general, key changes, chromatic inflections, etc....are all absent in this kind of music. Which isn't to say anything bad about modal music, just that it's much simpler. Because of the added complexity of diatonic harmony, there are many many more ways the music could "go wrong" so to speak. Most people are so grounded in diatonic harmony that they would easily perceive even small mistakes (or statistically speaking, deviations from the norm) without necessarily being able to explain what rule, exactly, is being broken.
It's also monophonic music; there's just the one voice and no accompaniment (other than a completely static rhythmic accompaniment that was added to the performance).
Finally, even within this much simpler framework, I'd argue this tune gets it wrong in a big way: it doesn't know how to come to an end. It just sort of stops, in medias res. In a lot of folk music, you'll find that the way a tune is ended still tends to hearken back to diatonic harmony: some sort of motion from dominant (e), maybe even with a raised 7th scale degree to tonic (a) that's outlined by the melody. That doesn't happen here, which is why the tune sounds like it just got cut off.
I find these NN experiments in music generation quite interesting conceptually, but so far the results--as music--have been pretty disappointing. I suspect that you could actually build a model that would allow for algorithmic generation of folk tunes that would produce music that would probably be more satisfying. The number of rules that govern a lot of kinds of folk music are small enough that you could encode many or at least most of them in your model. [1] However, at the end of the day you'd still just have a model that would only generate a fairly limited spectrum of folk music--say Irish gigues, reels, hornpipes, etc--whereas the dream with NNs, markov models and other statistical methods is that you could plug in any corpus of songs without understanding a thing about their harmony, form, structure, melodic patterns, etc. and get back music that sounds the same.
[1] and this is really massively simplifying on my part w/r/t the varied amount of folk music out there, some of it quite complex