My husband is under the impression that he can keep a big pot of soup, or a stew, out of the fridge and on the stovetop ”indefinitely” as long as he boils it once a day. I think he's crazy; inviting all sorts of nastiness to grow in there. Who is right?

Hot food hot, cold food cold. In between is the danger zone. It needs to be refrigerated. Heat can keep bad things from growing stuff, but when it is in the in between temps the pathogens can multiply, the heat won't destroy them.

Here is a good link to pass on to him.