I'll make this more explicit, and note that at present I don't see it affecting anyone's argument:

ZF can't prove its own self-consistency and thus can't prove it has a model. You logician says that's as it should be, since not every model of ZF contains another model - which seems true provided we treat each model more as an interpretation than as an object in a larger universe - and then says that a model has to be "well-populated" - which AFAICT holds for standard models, but not for all other interpretations like those your logician used in the previous sentence.