I am struggling with why the practice of storing HTML partials in SQL is necessarily bad.

If anything, this is arguably one of the more structured ways you could go about attacking the problem of a complex web product.

Because you shouldn't repeat yourself. If you're going to store your content that way, it should be as agnostic as possible, so that when the inevitable redesign comes around, you're not stuck rewriting every single line of that database, like I was.