I know that this sounds like a value judgement about the "importance" of languages, but hear me out. One of the most important recruiting grounds for the Eastern Roman Empire was the Latin-speaking lands of what is today Romania. What we typically see in the Roman Empire is that lands that experience thorough military recruitment also experience extensive Romanization, with the exceptions of Greek territories for special reasons, and the Middle East in which Roman rule wasn't quite as firm as it was in, say, Spain. Something we also notice about the languages not supplanted by Latin is that they're usually heavily attested, have strong literary traditions and are more or less historically "significant" languages. Historically "insignificant" languages, i.e. languages in Roman territory without strong literary traditions, were mostly fully supplanted by Latin, with the notable exceptions of Basque in Iberia and the various Celtic languages of Wales and Cornwall. However, we can attribute the survival of Basque to the relative isolation of the Basque country, and the far away, difficult to reach location of Wales and Cornwall. We can't make this assumption for Illyria. The Illyrians, unlike the Basques and Welsh, were within spitting distance of Italy and provided a massive manpower sink for both the Western and Eastern Roman Empires. Rome experienced a huge number of provincial, Illyrian-born emperors. Why is it then that Albania, a land close to Italy that was hugely important to the Romans and well integrated into the Roman Empire, managed to maintain a thriving non-Romance language while remaining very poorly attested until the Middle Ages with next to now literary tradition to speak of?
I don't mean to insult, but this is an incredibly myopic view of both language and history and makes many assumptions that are simply not the case. Ethnolinguistic populations expand and contract on their own due to more factors than occupying forces, and there is no explaining any of this in terms of 'significance'. I also don't see how a culture has to be a literate one to be significant. You're clearly working backwards from the vantage point of which cultures retained a language distinctive from their occupying forces, and trying to come up with underlying commonalities for that being the case, but 'significance' is not a cause; it's a result, and an often arbitrary one.
First, in case this was implied, and I can't tell that it is, there is insufficient evidence linking Illyrians and Albanians as one people. There is insufficient evidence to suggest the Illyrians even were one people. It is a possibility the Romans used it as a catch-all for foreign populations in the eastern Adriatic area, and that can't be discounted. According to autosomal analysis, the Albanian-speaking population was quite small in the past, experiencing rapid diversity only in the last 1500 years. This gives credence to the theory that Ptolemy's so-called Albinoi were either a small, distinct people, or a different people altogether. The name would then be geographically meaningful, but not ethnolinguistically meaningful, in the time between Ptolemy's attestation and early Roman ones. The earliest concrete references to the people we know of as Albanians are from 11th century Byzantine authors, like Attaliates, much later. As a linguist I consider it highly unlikely the 'Illyrians' in Rome's army were all speakers of Albanian or a predecessor language, or Albanian loans would be common throughout the Balkans at the least, just as Mongolian invasions disseminated loans from conscripted Turkic peoples, not Mongolic languages. Note that I say loans, not language.
I'm also curious about your sources saying Albania was "a land close to Italy that was hugely important to the Romans and well integrated into the Roman Empire". Integrated in what way?
I also have to take issue with the idea that languages were supplanted by Latin at all. Which languages are you thinking of? Roman occupation was not necessarily one of cultural assimilation. Administrative centers would use Latin, but the surrounding areas spoke native languages, like Gaulish or Iberian - both of which had writing systems, incidentally. Eventually factors like population expansion from Latin-speaking cities, or a cultural or religious connection to Rome, are what led to these languages finally dying out. Therefore if there are no Roman settlers living among Albanians, or their antecedents, the language won't be lost. See this thread and answer by u/rusoved. It should also be said that Latin wasn't even uniformly spoken throughout the Italian peninsula. There are dozens of unique Italic languages attested such as Oscan, Umbrian, and Venetic which persisted into the first and second centuries CE. Roman troops would have had a variety of native languages, but the troops' speech wasn't what determined the linguistic evolution of a region they occupied. Only sustained civillian generational contact can do that.
Madgearu, A. (2008). The wars of the Balkan Peninsula: Their medieval origins. Lanham (Md.): Scarecrow Press.
Ralph, Peter and Graham Coop (2013). The Geography of Recent Genetic Ancestry across Europe. PLOS Biology, v. 11, issue 5. May 2013.