Actually I'd argue it is the government's job to protect cultural value; that's precisely why they fund libraries and museums.
No..... libraries and museums are common goods which the public wants and everyone benefits equally from, and preservation of cultural history is one of the benefits. It is the job of the government to support such common goods, as long as there is majority support for the good.
Without government support, then there would be the problem of freeriders --- people who paid nothing in the long run, would get just as much benefit from the existence of the good as those who did pay for the construction of the library or museum.
It's the central purpose of government to provide a structure to help fund such goods, by requiring a majority to agree --- then everyone has to pay their equal share (relative to the benefit they and their descendants will derive from that good over their lifetimes), no freeriders, no tragedy of the commons.