... you are absolutely correct that there never was a Judeo-Christian tradition and never will be because foundations of the two are absolutely incompatible, while many today's values predate Judaism, all those Mel Brooks', pardon me 15 Commandments, by many millennia and are the result of the developing of the socio-centric behavioral pattern in ancient human societies where the survival of the community depended on cooperation, on family for procreation et al--thus first "values" and morals emerged.
In terms of Islam--the presence of Prophet Issa (Jesus) in Islam and Jesus kicking evil's ass just to establish Sharia Law is also not a theology one wants to draw parallels to. Not to mention the fact that the preservation of the Christian Holy Places by Muslims was primarily done during the reign of first four Caliphs, who were "liberal" and allowed a degree of religious "free" thought until Al- Ghazali wrote his main "work" Tahāfut al-Falāsifa, the Asharites won over Mutazilites and any discussion of the nature of Allah and Quran became a biggest no-no and Sunni realm lives in deep sewer ever since, especially in the last 500 years when it was ultimately beaten back by the combined West and then subjugated due to its backwardness, which persists even today. And in terms of preservation--it used to be Hagia Sophia. Not anymore--it is Hagia Sophia Grand Mosque now--a symbol of the conquest.
In related news, apostasy in Islam requires a good level of conspiracy skills and cosmetic surgery, because you could be killed for that--so run, and hide. Many were, when they wanted to quit Islam. So, neither Judaism nor Islam are compatible on a fundamental level with Christianity, not to be mistaken with evangelical death cults calling themselves "Christian", in reality being an outgrowth of Judaism, in the US. But in general, Europe is post-Christian both in (lack of) faith and cultural tradition and the same goes for the US. Europe is being Islamized as we speak and France is the first candidate on final "cultural enrichment" and disintegration, UK is next. And yes, Judeo-Christian Tradition is a sheer BS created after WW II for political and ideological reasons.