From what I understand, this shift from the "original" Jewish bloodline to the Khzars happened around the 8th or 9th century.

If so, I think having 1,000+ years history of being Jewish/claiming Jewish/whatever gives them the right to whatever religious/cultural/ethnic identities they now claim.

I honestly don't understand this argument.

It seems as if you're saying that because they don't have the same genetic profile as ancient Israelites, they aren't legitimately Jewish?

To be clear, I don't think "original" Jews, or those whose genetics are from ancient Israelites are in any way special or are more deserving of having their religion/culture/ethnicity respected any more than anyone else is.

So, even if Khzars did convert over 1,000 years ago, well damn. I think they've been around long enough to be able to legitimately call themselves Jewish.

This argument makes zero sense to me. It would be like me saying that because white Christians of anglo-saxon descent aren't the same bloodline/DNA as the original biblical Christians, then they're not really Christians at all, except that they follow the religion and they use the Christian religion as a shield for the things they do.