This component is great, but I believe it lacks an important feature: the ability to move a link to the hamburger menu when, on mobile, it is hidden from the header. Indeed, in most cases, I think removing the link entirely breaks the UI.
Because I urgently needed this feature, I did a fork:
However, I would love to see this in the official component.
You can see a demo here (check this page both on desktop and mobile).
Thank you for sharing your work @syl 
The goals of this component are pretty clear, it allows you to add links to the header with as little work as possible. As such, there are no plans to move the desktop-only links to the hamburger menu on mobile.
We’re generally very careful not to introduce any breaking changes, so it’s definitely feasible for you to maintain a fork of the component, but it’s not something that I would recommend.
That said, if you’re OK with some links being in the hamburger menu on mobile, then it follows that said links are not of the utmost importance and wouldn’t need to be in the desktop header in the first place. Have you considered using the Custom Hamburger Menu Links component for these links alongside this component?
Such a setup would allow to you add the most important links in the header, while keeping less important ones tucked away in the hamburger menu on both desktop and mobile.