Such holy sites present a problem. That is, they are not owned as such, but they are controlled by foreign religions, and even foreign nations, who *assume* they are property, almost like embassies.
So this is not like eminent domain. Instead is is a claim that sites in Israel are in Israel, not the territory of someone else.
Imagine if the UN declared Yellowstone park to be a World Heritage Site, and then started setting up UN toll booths for Americans who wanted to visit it, demanding money and passports?
This is somewhat the current status quo in Israel with holy sites. That foreign religion and even countries are claiming to *own* part of Israel, usually to threaten Israel if they interfere with that religions or nations control of that site.
Your analogy is flawed. The Churches which have possession of the holy sites predate modern Israel by up to 1900 years in some cases. If the U.N. owned Yellowstone 1500 years before the United States was established they would have a valid claim to it.