Depends on the structure.

The Parthenon was actually fine up until the 17th/18th century. It got blown up because it was used to store gunpowder during a war.

The Colosseum got destroyed by an earthquake around the year 700. But by that time, the thing wasn't really used for anything anymore, and repairs would require a lot of money and resources that nobody wanted to contribute.... Because it had no used

The big thing to keep in mind is that the main reason these things are awe-inspiring is because they were so massive, and expensive to build. Most countries don't wanna pour billions or trillions of dollars into buildings what solely amount to tourist attractions. The colosseum is basically just a magnet to get foreigners to buy hats at this point.

Especially since the materials used to build them could go towards more practical uses, there's just not any good reason. They're expensive, and currently culturally irrelevant. (Italy isn't Rome. Modern Greece isn't ancient Greece)

Greece especially isn't in an economic state to make that decision. Most countries have other problems to tend to