Why is everyone saying this doesn't exist? It's right there on the linked page! It's called "Approved Content Only" and I assure you that it exists, it's a real feature, it works just like you want, I use it myself, my kids watch Primitive Technology and Smarter Every Day and they can't watch videos I don't whitelist.
It does have a few issues. It's not reliable in showing everything you allow, sometimes things are missing for no reason, other times it will prevent you from whitelisting a video because it contains product placement (why does Google get to decide that for me? I'm an adult and can choose what level of product placement is acceptable for my kids). But it is a true whitelist mode and won't show other videos, just as requested.
It does have a few issues. It's not reliable in showing everything you allow, sometimes things are missing for no reason, other times it will prevent you from whitelisting a video because it contains product placement (why does Google get to decide that for me? I'm an adult and can choose what level of product placement is acceptable for my kids). But it is a true whitelist mode and won't show other videos, just as requested.