Your seatbelt analogy isn't legit. Since not vaccinating your children isn't illegal, you could only compare it against something that is also legal, whereas since not wearing your seatbelt is already illegal. Therefore your analogy is illogical because it relies on the belief that vaccinations should be mandatory to be true (which is the core topic of the debate). Since your analogy tries to support that belief, it doesn't really work (think the bible is true because it says so in the bible sort of situation).
However I have to agree with you in a way, because I believe it is the responsibility of the state to protect its people, in this case, from each other. But I think that the idea of the parents losing responsibility of their children to the state comes close to crossing the line of children being raised by the government. Should parents be punished for messing up (when not breaking the law)? I think not. Should vaccines be mandatory? Perhaps so.