As you further explained your question in various comments, you seem to be asking why the Bishop (Patriarch) of Jerusalem is Greek, rather than "someone who is ethnically Jewish, and practices Judaism along with their Christianity."
I am not sure there is a clearer answer to why "someone who is ethnically Jewish" is not the Patriarch of Jerusalem, than the vast majority of Orthodox Christians living within the boundaries of the Patriarchate of Jerusalem are ethnically Arab (generally Palestinian) and not Jewish. Perhaps there are Jewish converts from time to time, but they seem to be very, very few. As you probably know, Judaism was virtually wiped out from the Roman Province of Judea by the Emperor Hadrian after the Bar Kokhba revolt in the early second century. The Jewish population of Palestine waxed and waned for centuries, but by the time the Ottoman Empire was dismembered after World War I there were no more than 10,000 Jews in the area.1 In the mid-19th century there were about 60,000 Arab Christians in Palestine, and about half that number of Jews.2 Today the percentages have more than inverted, with only about 2% of Israel being Christian, 14% Muslim, and 81% Jewish. Today only a single majority Christian village (Taybeh) is left in the territory of Israel, Gaza, and the West Bank.
The above doesn't explain why the Patriarch is ethnically Greek instead of Arab (as is the case with the Patriarchate of Antioch), but it does help explain why he is not ethnically Jewish.
As far as why the Patriarch of Jerusalem is not someone who "practices Judaism along with their Christianity" the answer is much clearer: because it has been expressly forbidden by the canons of the Church dating back at least 1,600 years. For example:
Canon LXIV of the Apostolic Canons (dating to sometime prior to the 4th or 5th centuries) reads: "If any clergyman or layman shall enter into a synagogue of Jews or heretics to pray, let the former be deposed and let the latter be excommunicated."
Canon VII of the 3rd Ecumenical Council at Ephesus (431) reads "those who shall dare to compose a different faith, or to introduce or offer it to persons desiring to turn to the acknowledgment of the truth, whether from Heathenism or from Judaism, or from any heresy whatsoever, shall be deposed, if they be bishops or clergymen; bishops from the episcopate and clergymen from the clergy; and if they be laymen, they shall be anathematized."
Canon XI of the Quinisext Council at Trullo (692) reads: "Let no one in the priestly order nor any layman eat the unleavened bread of the Jews, nor have any familiar intercourse with them, nor summon them in illness, nor receive medicines from them, nor bathe with them; but if anyone shall take in hand to do so, if he is a cleric, let him be deposed, but if a layman let him be cut off."
Canon VII of the 7th Ecumenical Council at Nicaea (787) reads: "Since certain, erring in the superstitions of the Hebrews, have thought to mock at Christ our God, and feigning to be converted to the religion of Christ do deny him, and in private and secretly keep the Sabbath and observe other Jewish customs, we decree that such persons be not received to communion, nor to prayers, nor into the Church."
1. see, e.g., F. Peters, The Monotheists: Jews, Christians, and Muslims in Conflict and Competition, Volume II: The Words and Will of God, p.287
2. C. Rubenberg, Israel and the American National Interest: A Critical Examination, p.26