In Israel, it is forbidden to sell DNA kits. The official explanation is that the establishment doesn't want people to find that they might be sons or daughters of someone who is not a Jew and that will risk their option to get married as a Jew under the religious authority in the country.
Here is a translation from an article in an Israeli newspaper that talks about the topic:
And if we find out that we are not Jewish? "Look for wings, not roots"
Another concern is a sensational discovery that some of us may discover: maybe we are not Jews? Beyond the private shock (for those who care about it) and the consequences for rabbinic marriage (if the results are revealed) does this have national consequences? Most of the YouTube videos on the subject are entertaining, but there are a minority who find out that they are adopted, or that their father is not their biological father.
We asked Prof. Shlomo Zand, who wrote a book on the origin of the Jews and concluded that there is no such thing as a Jew, not according to common DNA in any case, what he thinks about the phenomenon and the prohibition that exists on private DNA tests in Israel. In addition to the fear of discovering bastards, is there also a fear of discovering that some of us are not Jewish according to DNA? Is there even Jewish DNA? Does the younger generation care at all?
Prof. Shlomo Zand ()
Prof. Shlomo Zand. If the Jewish people have no connection to Israel - that's not a bad thing
"At Vishva University they repeatedly publish articles about the existence of Jewish DNA. I think they shoot first and then mark the target," he claims. "All in all, this matter of DNA is in its infancy. The discovery of the structure of DNA has led to tremendous progress in the criminal field, it is a major player in the search and detection of hereditary diseases, but determining whether someone is Ashkenazi based on DNA is stupid, because for example when I got married I was asked to undergo a carrier test for Ty Sachs. They discovered that even non-Jews from certain regions in Poland are carriers of Tai Sachs."
But Polish Jews don't always look like Poles. And not necessarily like Khazar-Turks as you claim.
"I am a historian. I was a professor for 32 years at Tel Aviv University and I do not know a single book about the exile of the Jewish people. There are hundreds of books about several streets in Warsaw, but about the act of emigration from Eretz Yisrael - there is not a single one. So did we come from the Land of Israel and wander among the nations? probably not. I know, I'm considered a troublemaker."
I assume that many order the kits for fun, not always to make sure they are Jewish
"If it's for a game - I don't care. Curiosity is good. If these are really people looking for an identity, I recommend looking for tomorrow and not yesterday. As long as you don't develop 'nation' and 'race' according to that, it's fine. There is a fear that we will find out that we have no genetic connection to the Land of Israel."
What do we care? Do Australians have a genetic link to Australia? No and yet no one disputes their presence there because of that.
"Exactly, our existence here is not due to DNA, not because we were discovered and returned. But for some reason this has become a historical fact that justifies our existence here.
Source: Ynet.co.il
There is no proof or reliable source in the above article that connects the current Jews with the ancient Hebrew people. There IS emphasis on the DNA connection between the Ashkenazi Jews and themselves as a group, which is not hard to imagine as this group of people has been marrying only people from the same group which originated from Kazaria, which is today's area of Ukraine. So, there is what they call Ashkenazi DNA that relates them all but there is NO Hebrew DNA that they carry.
Here is a very interesting article about the real origins of the Ashkenazi Jews (Khazarian): https://www.freepolitik.com/p/demolishing-the-myth-of-modern-zionism-342
