That a pretty modern understanding of Judaism, historical there have been a lot of Jews without Jewish ancestry, but because of the stop in conversion and the practice of inmarriage have meant that most modern Jews have diaspora ancestry. To use a relative modern example Ariel Sharon’s mother belonged to a community of Russian converts to Judaism with no known Jewish ancestry.
Well, I was under impression that Ariel Sharon belongs to the modern history.

His mother was, indeed, from the sect of Subbotniks but they seemingly appeared in the XVIII century during the reign of Catherine II and were originally Christian peasants of the
Russian Orthodox Church. Not sure that your idea can be transferred retroactively to the VIII century (as I said, religions are not my are of interest). The same (at least as one of my friends assures me

) applies to the "Jewishness by mother's line" which (according to him) was adopted in the XVII century after the Cossack Wars, which made it close to impossible to tell anything definite about person's father.
Anyway, individual conversions of the non-Jews had been definitely happening.
If the Khazar convert to Judaism it would almost centainly be recognized by other Jews. The main problem may be that the Khazars may very well have been Karaite Jews, which could have lead to Rabbinic Jews not recognize them. In fact a interesting idea could have been if Karaite Judaism would have spread the Turks of Central Asia, resulting in it separating from Rabbinic Judaism and ending up being seen as the fourth big Abrahamic religion.
OK, as far as I can say, there are 3 main views on the whole issue (as I said, I have no opinion of my own):
1. Did not happen at all: no archaeological evidence and the documents are of a questionable truthfulness.
2. Did happen for ALL Khazars, which is Koestler's theory (AFAIK, there is a considerable criticism of it) and which, as I understand, implies that the Khazars too claimed the status of a Chosen Race, who made their own Covenant with the LORD, even though
they were not descended from Abraham's seed (
http://www.hope-of-israel.org/KhazarConversion.html)
3. Was limited to the leadership (seemingly adopting Talmudic Judaism)
Option #2 assumes a national level conversion which probably assumes, in practical terms, that if somebody does not consider them being Jews, this is his problem, not their.