I'm not making either of those claims. The first is an article of faith, and the second is trivially true.
I'm stating that one can't claim a particular set of DNA patterns makes someone "Jewish", because DNA is a biological phenomenon, and ethnicity/culture/religion are cultural phenomena.
DNA patterns often reflect the history of rigid social organization patterns like tribes (and indeed are essential for genetic risk management provided by organizations like Dor Yeshorim).
But nobody was tracing ancestry using DNA until the latter half of the 20th century, so claiming DNA as a basis of tribal identity seems quite anachronistic.
No that's not what I said. I said that DNA, though inherited, is irrelevant for determining cultural identity.
If I discover tomorrow that I carry YDNA haplotype J1, that doesn't make me Jewish, either culturally or ancestrally. Nobody could claim me as ancestrally Jewish based on that either.
Having a documented lineage stretching back many generations, however, would possibly do both, should I choose to embrace it.
Then substitute it with haplotype G or E1b, or include all of them. Or swap out both the haplotype and the cultural group for different ones, i.e Y Haplogroup R1b among people from Iceland.
The particular haplotype is irrelevant to the argument, which is that genes don't define cultural identity, and at best are trailing and largely low resolution indicators of historical social patterns.