I imagine that primer (plain gray spray primer) would be a good idea, though I don't think that the originals had that. Satin black (for steam) hardware store paint is indeed a good choice for this. Those old steamers were household appliances more than models.
Buy a couple of brands and do a test shot or two and compare. I personally do not like the now too-common "2X coverage" paints - poor coat control. Good for lawn furniture, maybe. Several thin coats is the way to go. No hurry.