I wouldn't say it was stagnant per se. I would instead say that the whole notion of "progress" in music just doesn't provide a good picture of what was going on. Progress from what, toward what, for what reason? Much of the driver of what we think of as modern-day music theory has to do with matters of notation, and what purpose does notation serve? To preserve, to remind, to communicate over vast distances. These issues are important if, say, you are the Carolingian Empire looking to standardize aspects of the Christian faith. But if you aren't interested in standardizing musical practices, then what use would a widely adopted notational system have served? Musicians have always gotten along perfectly well without notational systems (and continue to do so today, there are many great musicians who learn without recourse to musical notation). So what need would have driven it?

But music theory also has its speculative components too, especially for Greek culture. The history of why Greek thoughts about speculative music theory take root in Roman culture would be intimately bound up with the broader question of why Greek philosophy infiltrated Roman thought to the degree that it did. Why does Cicero care so much about Stoicism and Epicurianism? Why does Boethius have such a Platonic and Pythagorean bent? These are influences in general, that happen to manifest musically, but also have a wider history as well. (I wish I knew more about this subject, but I don't, sorry!)

Speaking of Boethius, it is really hard to say that Rome contributed little to music when Boethius's De institutione musica of c. 510 is essentially the foundational work upon which the very notion of music theory as we know it today was built, and was especially influential for Carolingian thinkers in the 10th century, such as the anonymous writer of the Musica enchiriadis and Guido of Arezzo. It does riff considerably on earlier work by Ptolmy (Harmonics, sometime in the 2nd century), which in turn was developing Pythagorean ideas, though. But regardless, it is certainly worth noting that it is the ideas of a Latin author that works its way into the bones of western music theory.

Lastly, an essential aspect of Roman culture was its cosmopolitanism. Rome was a massive empire, and as a result was the site of massive migration (whether voluntary, slave-based, or legally enforced) and mixing of cultures from various parts of the world (a great monograph on this subject, though one that unfortunately does not discuss music, is Tacoma's Moving Romans). This is certainly felt in the musical domain as well. Roman culture brought together and developed musical technologies from around the world. From central Italy came "native" players of brass instruments: the tibia, the cornu, and the tuba. Likewise, the Hydraulis was brought from Greece and is the forerunner of the modern organ. From Egypt came the cult of Isis and the rattle-like sistrum. The tympana and cymbala came from Asia minor. And so on. There are so many mingling musical traditions in the Roman Empire that often get submerged under the presumption of a predominantly Greek-based influence.

TL:DR. To say that Roman music is all just copying the Greeks is sort of like saying all of American music today is just a derivative of the Blues. Yeah, the blues has had an overwhelming influence on contemporary musical practices, but there's so much else that has also shaped what we listen to. It happens to be the case that the main contribution of Rome to the history of musical ideas and music theory is one person's interpretation of several centuries of Greek and Roman thinking about a Greek-originating musical philosophy. But don't let that fool you into thinking that the Romans did nothing but copy the Greeks in all their musical activity.