What trim are you/he referring to? The scotia/cove molding under the nose of the tread?.
IF that's the trim you are referring to, I agree that it is very difficult to refinish well. I've done it but it's time consuming and the end results were less than stellar. In all likelyhood, replacing it will be your best option. You could try to remove it and refinish it with stripper and such but what's your time worth? That trim doesn't cost that much. If you are painting the risers, you could leave it in place and paint the trim as well, although typically, the scotia molding is the color of the treads, like this.