A rather large part of the reason those shots cost $300 - as you say, after the insurance had paid part of it - is the very economic structure that pays for them. A recent government estimate is that it costs $2,000 or so for all the shots a child should get up to age 18. This is a predictable expense of having a child. There's simply no insurance model that is going to lower this cost in total. It's just not a low probability event that can actually be insured. We can have assurance for it, sure we can, which is rather the point of the article.

Please do note, I'm not insisting that parents must pay out of pocket either. I'd be just fine with the general taxpayer paying for all vaccinations to be free at the point of use. But "insurance" still isn't the right model.