Mine are older (5 and 7 yo) and trough down loads of snacks between meals and still eat more or less whatever is put in front of them.
They are tall, healthy and slim too
I think the no sugar idea is very bad for children, I firmly believe that their bodies need sugar to get them through the intense energy demands placed on it.
I loathe aspartime, sorbitol and other artificial sweeteners with a passion and mine always have full sugar versions of things. However, they are very good at moderating their intake, helped a little by me
They do get a bit of choice as to what snacks they choose, afterall we are trying to introduce the concept of selfcontrol here rather than parental control. Over Christmas, despite chocolate and other sweets being freely available, they've eaten their own weight in clementines!