Add support for PRIVATE header sets, analogous to the existing
INTERFACE header sets support. A key difference between the two is
the way they handle build and usage requirements. INTERFACE
header sets rely on the original target's INTERFACE, whereas PRIVATE
header sets copy across various properties from the original target.
Generated files need special handling for PRIVATE header sets.
Add a dependency on the original target to the verification target to
ensure such generated files will exist.
Fixes: #23448
