It's nothing to do with the effort involved. One airline is simply not going to agree to take your bag and later hand it over to another airline if there isn't a contract between them specifying who is responsible for what, and who is liable for what if something goes wrong. If these airlines don't have such an agreement between them, then that's that.

However, I am surprised that AF and NZ don't have an agreement. I have a suspicion that you've actually hit a different problem: Many airlines will no longer transfer bags to another airline if the travel is written on two separate tickets (as yours is), as strictly speaking there has never been any obligation to do so.

If you wanted collaboration, you would have done better to ensure that your flights were with airlines in the same alliance. They will usually transfer bags to other alliance partners even if there are two separate tickets involved. I suspect that there are several options that you could have taken that would have been all within the same alliance.