Are you asking "Why do Christians in the Middle east" not attempt to claim Jerusalem?
Or simply Christians, in general?
Christians make up 2% of Jerusalem's population. And that 2% carries over to Israel, overall. For the Middle East as a whole, they seem to average ~4% of the population. Outliers like Syria (10%) and Egypt (6%).
PBS Global Connections: the Middle East
I would say this marginal demographic regarding Christians has a lot to do with the lack of any claim on Jerusalem from within the Middle East, itself.
There is no geographical contiguousness for Christianity in the region. The nearest Christian-majority nation (by land) to Jerusalem is Armenia -- and between the two lies a sea of Islam. It would make no sense for Christians to claim a city which would produce such a cultural and demographic island-outlier. Israel's existence (right or wrong) at least provides a geopolitical grounding in the region for Judaism. Islam's overwhelming dominance in the region today, combined with ~900 years of momentum in the region historically, provide its grounds for a claim.
While Jerusalem played a large role in the minds of Medieval Europeans -- some of the Crusades which centered around conquering the Holy Land, obviously, as well as Jerusalem as a nodal point on Medieval maps, which were centered around the city -- Western Europe only actually controlled the city sporadically. Byzantium (now mostly Turkey, which is noteworthy here) lost its dominance of Jerusalem in the 7th century. For the next 400 years, it is within various caliphates. In 1100, it is conquered by Crusaders and held for less than a century. It is very obviously a foreign occupation and the Crusader States are in a near-constant state of besiege. No lasting historical or political anchors are put down, and Jerusalem naturally reverts back to the Ummah when Saladin recaptures it. Except for a brief period in the 13th century, Jerusalem is now under constant Muslim control, up until World War I.
Christians have a better claim on Istanbul than Jerusalem. Its historical legacy there is more recent, and it at least is neighbored by Christian-majority nations. And only extremists call for the reversion of Istanbul to Christendom.
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