I'm not really familiar with the modern Assyrian ethnic group, but will point out that anyone tracing their roots to iron age Mesopotamia probably has plenty of ancestors living in both Babylonia and Assyria. Ethnic groups aren't as simple as that, and modern Assyrians' identification with Assyria specifically I assume is more complicated and someone more knowledgeable can answer!

Anyways, u/Bentresh has a good answer to the question of Aramaic spread here, and in their other answer they link to: https://www.reddit.com/r/AskHistorians/comments/ofkl5a/when_did_ethnic_assyrians_switch_from_speaking_an/

Aramaic became the lingua franca of the neo-Assyrian empire since a lot of people spoke it, and the empire forcefully deported a lot of those people all over the empire, spreading it everywhere. And it's really difficult to overstate the scale of those deportations, this is millions of people over a few hundred years.

It didn't actually "replace" Assyrian in the sense that if you were writing a letter to the King of Assyria you had better use Assyrian, as you'll see in an anecdote in the answer. In that sense it wasn't prestigious; it was the language of the common person, widely used and understood across a large geographic area. The Achaemenids didn't "make" Aramaic the official language. They didn't really have something like that - the Persian people, including the kings, probably spoke Old Persian, and major inscriptions often included multiple languages (such as the Behistun inscription).

The Syriac Orthodox Christians are descendants from the Arameans that lived in what is now Modern Syria, South of Turkey, and other surrounding areas. They never claimed to be descendants of the Old Assyrians Empire but some members beleive in the new political and nationalistic Assyrian movement that was brought to life by British Oriantalist and British Foreign Service officers in the 19th century. Up until today, I haven't seen any scintific research that the Nestorians (Church of the East followers) who lived in the Hikari Mountains, Northern Iraq, around Urmia in Iran and some other pockets in Turkey before WWI and Iraq are descendants of the Old Assyrian Empire. It's possible that the Old Assyrian populations left their 3 capitals after they were sacked and destroyed by the Babylonian Army and their allies and lived elsewhere but nothings is proven yet. If you have sources, please share.

As for the Akkadian language it looks like it disappeared after the fall of the old Assyrian Empire. The new Assyrian language that is spoken by Church of the East and the Chaldean Christians originated from the Aramiac language and it's known today as the East Syriac to distinguish it from the West Syriac that is spoken by the Syriac Orthodox Christians. There are few other derivatives from the Aramiac language but it's getting coplicated here.

A good source for infomration about the Aramiacs and the Syriacs people, culture, and language is a book titled The Syriac World