Well if you ask the modern Assyrians, they'll proudly tell you that they do.

From a historical perspective, talking about "direct and distinct" lineages is a fraught topic. In a 2600 year period, it's basically impossible for anybody's lineage to be all that direct. I like to point to this study, which suggests that (mathematically) the most recent shared ancestor of every single European lived as recently as 1000 years ago. In a relatively short time frame, people wind up fairly intermingled. 2600 years is also a long time for any kind of cultural traits to remain distinct. A reasonable comparison might be Rome. How much do modern citizens of Rome have in common with the people of the ancient Roman Kingdom? In the last 2600 years, Rome has been conquered and occupied by significantly fewer groups than Assyria, and even spent most of that span ruled by institutions with roots in the ancient period.

What we can say is that the region of upper Mesopotamia was home to vassal kingdoms in the Nineveh Plains the ruins of Nineveh and Assur was still known as "Asoristan" by the Sassanid Persian Empire into the 7th Century CE. That was not a revival. It was "Athura" under the earlier Achaemenid Persians and Assyria to the Greeks throughout the 400-500 years between the fall of the Neo-Assyrian Empire in 609 and the first uses of "Asoristan" by the Parthian Empire.

Important cities and vassal kingdoms in that region, like Hatra or Adiabene, were hold outs where traditional Semitic paganism was still practiced alongside Christianity at least into the 3rd Century CE. In the 2nd Century, the pagan author Lucian and the Christian theologican Tatian both self-identified as "Assyrian." Some modern authors often inaccurately identify them as "Syrian" on the presumption that they were using "Assyrian" in the more archaic Greek sense of "anyone from east of the Euphrates. Seeing as both men were from the area of Assyria proper in Upper Mesopotamia, it seems more likely that we should take them at their word.

Around the same time, the city Assur was revitalized and the major temple of the eponymous god was rebuilt. The city flourished until it was sacked by Shah Shapur I in 240 as part of his reconquest of Mesopotamia. That's one of the last major moments of prominence for obviously "Ancient Assyrian" culture in antiquity, but it overlaps neatly with the first generation of prominent Assyrian Christians like Tatian.

Beyond that point, the Assyrian identity is harder to track. They were not the only Syriac-Aramaic speaking ethnic group to embrace Nestorian Christianity and schism with the Roman church, and thus get subsumed into a wider idea of "Syriac Christianity" in the western literature. The geographic label Asoristan remained consistent until the provincial borders of the Sassanid Empire were re-organized by the early Caliphates and Assyria was included in the larger province of Al-Jazira.

For about 1000 years after that point, it is very hard to track any clear references to an Assyrian ethnic minority. That doesn't mean all that much. The same could be said of most ethnic minorities in and around the area we could describe as Kurdistan today. That's not to say that the identity did not exist, just that it was not often documented as "Assyrian" because the Assyrians were often subject to the same historical events and prejudices in the Muslim world as other Christians who are also included in the broad ideas of the "Church in the East" or "Syriac Christianity." As Islam spread, that religious distinction helped form a tenant of group identity for some Christian minority groups living in the Nineveh Plains and the surrounding regions.

The use of the name "Assyrian" resurfaces in De dogmatibus Chaldaeorum disputatio, a 1617 book about a schism in the Church of the East, in which much of the community west of the Tigris entered into official Communion with Catholic Church and became the "Chaldean Catholic Church" in 1552. De dogmatibus Chaldaeorum disputatio contrasts that group with "Eastern Assyrians," who remained an independent denomination. There's no context for this reappearance or when exactly the archaic ethnic label of "Chaldean" was first applied to Eastern Christians. There are several books before 1617 that use Chaldean when describing the 1552 Schism, but no reference to "Assyrian" that I can find. It's an odd thing because no group nor place had been associated with a "Chaldean" identity since the 3rd Century BCE.

It was the Renaissance, a period of lots of archaic language in western literature and in how European writers talked about West Asia, but the choice is odd. The sudden, uncontextualized use of "Assyrian" in 1617 does suggest that the name was used to identify the Assyrian ethnicity/religion before that time, but I cannot find any specific example. It's entirely likely that it was simply the name Assyrians used for themselves and the patchy record of European contact with that region before the 18th Century just leaves a lot to be desired.

Since then, the historical record of an Assyrian ethnicity and the corresponding minority denomination of Christianity is consistent, due in part to increased European trade and travel with to the Ottoman and Persian Empires.

So while there is a lapse in documented use of the ethnonym "Assyrian" from 639-1617, all of the available information does indicate that there was a minority group in the Assyrian homeland with direct religious continuity from the pre-Islamic Assyrian Christian community. Nothing from the internal memory of the Assyrian community or outside sources after 1617 suggests that the name was applied to the Assyrians by outsiders, so it stands to reason that it was their own terminology. In lieu of any evidence for any medieval campaign against Assyrian identity, rather than Christianity overall, there is no reason to think that the name "Assyrian" was not used throughout the undocumented period as well.