Why DNA does not prove someone is Jewish just Common Ancestors of the Middle East aka Arabian Peninsula

 Pretty sure they already pointed out that originally the Jews were called Palestinians or Canaanites. Just as Syrians, Lebanese, and other Arabs also have ancestral lineage in the same area. Jewish religion is through the female line but the J2 haplogroup is male. European women married Jewish men. 

"the Phoenicians were Canaanites, justifying this based on their language (Canaanite) and the fact that “Phoenicians” called themselves Canaanites (Aubet 2001; Groot 2012; Padró 2012). It is thought that both Hebrews and Phoenicians were descendants of the Proto-Canaanites (Padró 2012)." 

So having that J Haplogroup doesn't prove it's a Jewish religious lineage. 

"Approximately 35% to 43% of Jewish men are in the paternal line known as haplogroup J and its sub-haplogroups. This haplogroup is particularly present in the Middle East and Southern Europe...

Then I discovered J2 is also very common among Iranians and Turks. So, what this tells me is it is not about language families, but about region. Middle Eastern people have a common origin." 

"Overall, NRY and mtDNA based motifs, previously reported to differentiate between groups, were found to be more represented in Jewish compared to non-Jewish groups. However, this seems to stem from common ancestors of Jewish lineages being rather recent respect to ancestors of non-Jewish lineages with the same “haplotype signatures. Moreover, the polyphyly of haplotypes which contain the proposed motifs and the misuse of constant mutation rates heavily affected previous attempts to correctly dating the origin of common ancestries. Accordingly, our results stress the limitations of using the above haplotype motifs as reliable Jewish ancestry predictors and show its inadequacy for forensic or genealogical purposes." 

Mitochondrial and Y chromosome haplotype motifs as diagnostic markers of Jewish ancestry: a reconsideration Front. Genet., 09 November 2014 Sec. Evolutionary and Population Genetics 

"To make an example, the “CMH” signature, in its classical and extended version, has been observed in many haplotypes of inhabitants of the Arabian Peninsula with typical Arabic names, as well as in many Jewish people belonging to haplogroups J1 and J2. "

 "An easier explanation is that, between 7600 and 10,400 years bp (95% CI), the “Cohen Modal Haplotype” was an ancestral haplotype for the historical inhabitants of the Arabian Peninsula. About 4000 ± 520 years ago the establishing Jewish population carried this “modal haplotype” along with the future Arabs, who at that time had a common ancestor with the future Jews (Klyosov, 2010). 

"By around the tenth century AD, a slightly modified “recent CMH” split from the “older CMH” (in more extended haplotype formats), while both of them contained the 6 marker signature of the “CMH,” which is still present in many Arabic haplotypes. This “recent CMH” became the ancestral haplotype for a separate albeit recent Jewish lineage within haplogroup J1. If one consider only “CMH” haplotypes within this population, a common ancestor who lived around 1255-986 years ago can be identified."

 "...the overall substantial polyphyletism as well as their systematic occurrence in non-Jewish groups highlights the lack of support for using them either as markers of Jewish ancestry or Biblical tales." 

"Then a split between the Jewish and the Arabic lineages in both J1 and J2 haplogroups occurred, which is clearly visible on the respective haplotype trees. The data show that a common ancestor of Cohanim (Jewish High Priests) of haplogroup J1 lived 1070+/-170 ybp, while a common ancestor of Cohanim in haplogroup J2 lived 3300+/-400 ybp." 

"But what about the CMH? Because the markers used to define it are of the fast-evolving type, the origin of this haplotype could be much more recent,...Therefore, if we accept the evidence from studies of the Y chromosome, showing the extended CMH in kohanim but not in Levites and in only a small percentage of Israelites, the biblical account of descent and relationships cannot be a true account. This result does not invalidate kohen status because kohen status, much less Jewish identity, does not depend exclusively on genetic makeup. Two respected geneticists, David Goldstein and Harry Ostrer, have each written books on the topic of Jewish identity in the DNA era. While recognizing distinctive patterns of genetic markers in Jewish populations, they concur on this. Goldstein states: “All I can do is echo what Tudor Parfitt has written: ‘Jewishness is not a matter of DNA.’”28