It would be nice if Cura would have a navigation cube known from many other 3D applications.

Nice one, @probonopd!
And if you're at it, add a possibility to enable/disable perspective (orthographic) view. This would make layer-view much easier to over view.
Unfortunately Autodesk has a patent on the navigation cube.
I'd be surprised if the general idea could be patented (might be wrong though, ianal). Do you have a link? Maybe we'd need to make it a bit different then, but still something in a similar spirit.
Use arrows and planes...
Actually if we could have this existing UI element interact with the mouse like a navigation cube, then that'd do the job:

Simplify3D has this. There the whole workspace behaves like a navigation cube.
@probonopd Software patents. Which would mean we can't distribute to america anymore.
Even if a court would deem it invalid, it would still cost a whole lot of money, so I'd rather not take the risk.
Let's consider the OP's last idea to circumvent the patent.
Caveat: for some printers, this is in the center of the buildplate.
It'd also be a bit weird for elliptical build plates to click on the sides.
This has been implemented with several buttons at the top of the screen in 3.1.
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Actually if we could have this existing UI element interact with the mouse like a navigation cube, then that'd do the job:
Simplify3D has this. There the whole workspace behaves like a navigation cube.