I don't offer my DCs snacks as a matter of course if they're at home with me (3.5DTs), and that's because when they were going through the fussy 2-3yo stage I realised they ate much better and tried more stuff when hungry. I know that's not true of every child.
However, on days at home with me they tend to have a late, large breakfast (porridge or cereal, and fruit, and toast, sometimes yoghurt or eggs), a decent lunch and a filling hot dinner. And we have the time to make sure it gets eaten.
On days when we're out and about, or they're at pre-school, or have friends around, or anything that disrupts the routine and means they are less likely to eat a proper meal, then we definitely offer snacks.
For me not snacking was about making sure more nutrition went in, not reducing the overall calorie count. And it's a very flexible rule - I will take them for cake straight after pre-school if I think they I deserve or need it 