I can totally see Steve being Superdad whenever JJ was't feeling well, out of sympathy for him. And maybe a kind of guilt, because obviously this is his fault for passing on sick genes.

Nora: He's got the flu, Steve. I've got the flu. Everyone has the flu except you and Bucky, because you can't have it. I don't think you can blame it on yourself.
Steve: I brought him some soup, and I put a fresh pillowcase on his pillow, and I got him a new book to read.
Nora: You're spoiling him.
Steve: I got you some flowers.
Nora: Oh. Well, that's all right then.