If it's super broken or critical, maybe a slot to let the user provide their desired header level would be an option? Or maybe even a simple prop would be enough. If your component is itself header-like, perhaps making 6 versions, fun-header-1, fun-header-2, etc. that are explicitly scoped to that level header would make sense too.

You could define a recursion depth parameter for your component. Or a specific header level parameter. I tend to think of spitting out HTML as something with recursive context. The CSS grid and even flex layouts come to mind as well.

Specifically, this is the document outline algorithm. But it’s not so much “not always implemented properly” as “not implemented at all”. No user agent has implemented it or seems likely to, and I don’t know of anything other than conformance checkers that implements it. https://github.com/whatwg/html/issues/83 shows a bit of the discussion that’s gone on; whether to remove it, simplify it drastically to the point that it’s actually sanely implementable, &c.