And that's the problem. Jews who abandon the Torah and who marry non-Jews are no longer true kosher Jews. And by your definition, the "original Jewish chrstians" would have been as Greek or Syriac as they were "Jewish."
I am just dealing in facts not emotions. Also "Jewish" is a word in English that denotes a religion and an ethnicity when the two are separated. Maybe Jewish Hebrews is a more accurate term. But it is clear Hellenized Jewish Hebrews converted to Christianity at a time when many of these very same Jews did not view it as a new religion but the Jewish religion with a Messiah. Over time it was differentiated as "Christianity" - a new religion.
In fact I may be so bold as to state the post temple rabbinic Jewish religion went through changes as well so it also is an offshoot of "Temple Judaism" and that eastern Christianity is senior to modern Rabbinc Judaism and that they should both be considered the offspring of the Temple Judaic faith.