I have three kids, two of whom are very picky. They used to eat a lot of their favorites and not much else but because I have an adventurous eater as well, I try to expand her palate whenever I can. I had the same approved menu as the OP when they were very young, over the course of a few years, I have immensely expanded their palate by not catering specifically to them. I always have one thing they will eat. I also have one thing they must try. It usually takes about 15 attempts with this extra food until they all request it but they eat some sushi, most meat, most soups and almost any kind of veggie except spinach. Along with the pre approved list of foods and it's quite a decent variety. I have them help me cook whenever I can and I also make sure I don't turn up my nose at anything either. Most of the success has been with the addition of food that other people enjoy. They see this and realize once they've tried it it really isn't that bad.
Ps- I haven't tried liver and onions. The one and only thing I Troy know I hate the smell and taste of!!