My kids have to ask for food. If I let them help themselves, even to so-called good choices, every apple in the fruit bowl would have a bite taken out of it. Not even mentioning the mess...

I think it is a bit much to want a 4 year old to make "healthy" food choices. I do give my children choices: do you want an apple or a banana. would you like peanut butter or cheese on your sandwich,... that type of thing. Not: have a look in the fridge and see what tickles your fancy.

I offer food at certain times during the day. Outside of that, they have to ask. Even my 1 1/2 year old comes to me and says: "apple" or "want biccie"

By the way, I don't care what nutritionists say. But there is no evidence that children who graze are worse of nutritionally or will become obese. It depends very much on WHAT they are grazing on. I believe in eating when you are hungry. Not just when some arbitrary time for eating has come. Of course within reason. But if my child is hungry half an hour before dinner, I see no problem giving them a healthy snack, like a piece of apple or a handful of nuts, to tide them over to dinner.
Same goes for breakfast. I know many people believe in eating straight away when they have gotten up. I find neither of my kids will eat more than a mouthful straight away. They usually refuse even that. They do much better with a slice of apple first, then have breakfast about an hour later.