That is a neat model. It seems like unlike printables, makersworld doesn't have a 3D preview on the website (or maybe you need to be logged in, or use an app or something)? As I have a Prusa printer, I have naturally falles into using printables (and Thingyverse back before printables was launched).
I will have to download the model and take a proper look in the slicer or CAD program when I'm no longer on my mobile phone. (I will probably not print it, as I'm not currently in the need of a desk organiser, and I don't like wasting plastic.)
For the benefit of those who don't know (I assume you do): Texture absolutely helps rigidity in any part, since if the surface is already curved or creased in one direction it resists bending in other directions. (This is why a rolled up paper is stiffer than a sheet.)
I looked at the model on printables, and have a followup question: why does the slit go all the way through? You can make the first few layers print normal and solid (in fact this is very common), so I'm slightly confused as to why you didn't (it would probably increase the strength a bit).
I have a long love affair with vase mode and abusing it, and always saving time/weight is my main goal, like in this other case: https://makerworld.com/it/models/840291-vase-mode-gear-phone...
Your blog post made me thing that we would almost need a specialized vase mode site for models of that kind :D Moreover, there is no reason why the top surface could not be closed with bridging. The slicers have a lot of odd limitations in the context of vase mode.
If you want full control though, you might want to look at https://github.com/FullControlXYZ/fullcontrol (using python to directly generate gcode). Perhaps a bit over the top, but their examples are cool and show things that can't currently be done any other way. You could definitely switch back and forth between vase and normal with it.
https://makerworld.com/it/models/99219-olivetti-style-vase-m...
You can create fast to print objects consuming very little filament, however to have some kind of texture on the surfaces is absolutely needed for strength.