Why add header files into the add_library/add_executable commands in CMake?

In our projects we use a “simple” way of yours – add_library with both headers and sources.

If you add only sources, then you won’t see headers in IDE-generated project.

However, when installing, we have to do it like that, using two install commands:

install(TARGETS library_name
        LIBRARY DESTINATION lib)

install(FILES ${PUBLIC_HEADERS} 
        DESTINATION include/library_name)

If you want to do it as a single command, you can use set_target_properties with PUBLIC_HEADER, as you suggested.
Then, this kind of install is possible:

install(TARGETS library_name
        LIBRARY DESTINATION lib
        PUBLIC_HEADER DESTINATION include/library_name)

Choose the one you like the most and stick to it.