I prefer to use FetchContent for dependency management.
With this:

FetchContent_Declare(
    zlib
    GIT_REPOSITORY "https://github.com/madler/zlib.git"
    FIND_PACKAGE_ARGS NAMES ZLIB
)
FetchContent_MakeAvailable(zlib)

the whole process of cloning the repo, generating and building it as a dependency might be automated by cmake.

The problem is that the example executable relies on the assumption that the whole project with the headers is the top-level cmake project.
And of course this is not true in case of FetchContent which will download zlib as a subdirectory.

There are several different approaches on how to fix this.
But the simplest one I can see is to add an option to disable tests and examples, like ZLIB_BUILD_TESTS.

I can provide an example project that uses zlib as 3rdparty via FetchContent to reproduce the issue if needed.