Why is ZF favoured over NBG

Solution 1:

You don't need to treat proper classes as objects in order to prove things in set theory. ZF is simpler, so it's the preferred choice. (EDIT: I can't speak to whether forcing is somehow simpler or more elegant in NBG, but this is admittedly the first time I've heard that claim. Either way, a proof in NBG is a proof in ZF, if you follow, so you don't need to say that a particular proof is "an NBG proof".)

To facilitate discussion about things which concern proper classes (such as the class of all ordinals), we can prove in ZF that the objects of a proper class exist and that they have some property, and then discuss the proper class as though it were an object in ZF. But it is understood that you could make this rigorous without the notion of "proper class" by referring to the property. We also use facts about proper classes that we can prove in ZF (again, without an actual definition of a class as an object). NBG just doesn't add anything useful to ZF that we can't abstract into a ZF discussion.