I think Duo knows the old ryhme, “Old Mother Hubbard, went to the cupboard, to fetch her poor dog a bone, And when she went (came/got) there, the cupboard was bare, and so the poor dog had none.”
We learned Kuhlshrank in school, which was fridge. Naturally DL now disowns it.
Every fridge is somehow also a cabinet/cupboard, but not every cabinet/cupboard is a fridge.
Jeder Kühlschrank ist irgendwie auch ein Schrank, aber nicht jeder Schrank ist ein Kühlschrank.
Should work just fine. When we lived in Germany "Schrank" was used for wardrobe/ free standing closet/armoire all the time. I think technically it may be "Kleiderschrank or Wandschrank" but I don't remember ever having heard people using that longer form when speaking colloquially. Although that may depend on the region. In the US they do not really use the (physical) free standing wardrobe/armoire ones, more the build in ones which are called a closet. Generally here, if you use wardrobe or armoire for an in build closet people look at you funny, if they even know what it means.