Getting a place in public primary schools for 5-year-old children
Elementary school starts at age 4 and is compulsory; before that, it’s childcare and you have to pay for it – the lottery element I guess indicates they have a waiting list, but school before age 4 is not obligatory.
I just did a quick Google search for “elementary school in haarlem netherlands” (link), I got a lot of hits – so there is no shortage of schools there. You won’t be able to register your child for any school until you have registered and got your BSN from the local council where you live.
When we moved there, we just registered, and walked over to our nearest school, helped the Headmaster climb down from the tree he had got himself stuck in, had a chat with him over a cup of coffee and then registered our 3 kids there. We never had any problems with the Dutch education system; the main issue that expats have is they expect it to be just like home, it isn’t.