The hamburger icon tends to be employed because designers—or more often, clients—aren’t fully committed to a mobile-first approach. They want a ‘regular’ site, but squeezed onto their granddaughters phone.
In short, the hamburger icon is a symptom of our collective failure to wholeheartedly embrace all aspects of the mobile-first approach.
In order to solve the hamburger issue, we have to accept that the Web as we know it doesn’t work. The rise of the mobile web means far more than reducing the number of columns we use and dropping some of the heavier image files.
The mobile web is used in a different way to the web of old. The mobile web exists in the context of dedicated apps, and users expect the Web to be experienced in a similar fashion.