I've realized that it's hard to make up categories and group links. Putting all links in a sort of 'grid' with some less strict grouping would work, but it would take up lots of space outside of a footer. But a footer would never work on some pages.
Putting such a 'grid' of links in a hidden box and having a single target which reveals the menu would defenitely work on all pages. On result pages and internal pages, the result page header could be kept, but the menu reduced to this single target.
If one wants to highlight something (like 'spread DDG' or 'settings'), its link could of course be added next to the menu target to make it more visible.
On the frontpage, the target could be put inside a small box at the top of the viewport:

It would be good to keep most of the links on the current front page because removing them would make it feel empty and unfamilliar. That bar of links helps users find the most important features and content and adding a link to it is a great way to announce things in a subtle way.