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.