- Levantines are descended from the Aramaic-speaking natives of the land, not from Arab immigrants.
- Levantines' language contains a lot that sounds like Aramaic.
- Therefore, Levantine is a continuation of Aramaic, not of Arabic.
Step 3, of course, does not follow from Steps 1 and 2. Step 1 is irrelevant to the whole question; the language of your ancestors is very often not the ancestor of your language (ask any Irishman, or any Egyptian). Step 2 is necessary but insufficient for getting to Step 3, since the statement is just as true of Classical Arabic - or of Akkadian, or Ethiopic - as it is of Levantine; we've already seen that deciding linguistic ancestry requires a more sophisticated toolkit.
Nevertheless, this impulse to emphasise continuity and downplay movement deserves more attention. In the Arabic-speaking world, the conspicuous problems with the existing political and economic order, and the humiliating contrasts between the ideals of pan-Arabism and the reality of closed borders and unchallenged occupations, provide an obvious local motivation to downplay Arab identity, and language is so central to pan-Arab identity that it could hardly be left unchallenged. But the impulse is not unique to the region; in some respects, it faithfully reflects wider intellectual trends of the late 20th/early 21st century.
During this era, immediately following some of the largest migrations and invasions in human history, many archeologists and historians have come to feel more and more uncomfortable with the very idea of either. Changes in material culture previously seen as the result of migration were re-explained as diffusion or independent innovation, and reports of barbarian invasions were reinterpreted or dismissed. In some ways, this has been a useful corrective to a previous era's overemphasis on migration; it has arguably made linguists more conscious of the familiar fact that language shift does not necessarily imply invasion, much less population replacement. In others, its influence has been rather less helpful. Linguists reached the late 20th century with a well-tested toolkit for studying the origins of basic vocabulary and morphology, its predictions spectacularly confirmed by such discoveries as laryngeals in Hittite and labiovelars in Mycenaean Greek. Applying this to most Old World languages, and many American or Australian ones, yields a story of discontinuity (be it through language shift or population replacement) that would be familiar to any 19th-century philologist, but that grates somewhat on postmodern ears. Of course, the same toolkit often allows us to detect substrata - elements left over from the population's previous language after they shifted to another one - but that's not enough to satisfy everybody.
A few linguists have responded by trying to change the rules of the game, insisting that the origins of a language should be determined not by vocabulary and morphology, as is normally done, but by purely structural features. This is an important component of Wexler's generally rejected claims that Yiddish is non-Germanic (and that Modern Hebrew is non-Semitic), and is the very essence of Lefebvre's somewhat more popular claims that Haitian Creole is just relexified Fongbe (and almost anything else with "relexification" in the title.) This approach runs into severe problems almost instantly - establishing the history of syntactic or semantic patterns is far more difficult than establishing the history of vocabulary or morphology, simply because the former are far less arbitrary and are chosen from a far smaller set of possibilities. To make matters worse, we also find major discontinuities in such patterns in cases where both the population and the vocabulary were relatively stable, such as the transition from Old English to Modern English. Johanna Nichols' efforts point towards the possibility of getting around this by identifying highly time-stable typological features, but the results, at their best, are not nearly fine-grained enough to support narratives of continuity in any specific location. "Continuitarians" in the Arab world apparently haven't gotten around to adopting this approach yet, except occasionally in Morocco, where academic linguistics is unusually advanced for the region; they surely will, however, when they realise that it could be extended to cases like Egypt, rather than being limited to the Fertile Crescent.
For much of the world, especially Europe, a complete lack of ancient written documentation makes another response available: simply argue that the language currently spoken there must have been spoken far earlier than previously assumed, and hence got there not through invasion but through some more peaceful process. This yields the various Paleolithic Continuity Hypotheses. The main problem with this for linguists is that it forces us to postulate a much lower rate of linguistic change for the past than is observed for languages with a long written history, or even for unwritten languages that happen to have been recorded as long intervals; as a result, these hypotheses have remained fairly unpopular. For the Middle East, however, the point is moot: writing has a longer history there than anywhere else on the planet, and that history reveals regular episodes of language extinction, language shift, invasion, migration, exile, and everything else that we're supposed to be de-emphasising.
So if you really want to emphasise your languages' continuity with your ancestors', these are two more promising ways to do it. But I would suggest that there's no reason to bother. If your current identity isn't working out for you, and you don't think you can reform it, why not work on creating a genuinely new one, rather than perpetuating the obsession with heritage by digging around in history for an even older one? It worked out pretty well for America, after all.