If this is "organizing things spatially on screen in a way that's meaningful to you" then I honestly don't know of a way of interacting with a computer that isn't that.
But the printer example is a great one. The value of printing the output is that then you can do things with it that help you think about it in ways that you couldn't do with the on screen interface. If you wanted to you could cut the paper up and rearrange it, draw diagrams etc.
There is a whole genre of interfaces, called Tangible User Interfaces or TUI, that is all about giving physical form to digital information and using physical objects / tokens to represent and modify information inside the computer. Here's a paper with an example task that's similar in many ways to the task in the paper in the original blog post.
https://www.cs.tufts.edu/~jacob/papers/chi02.pdf
So yeah, if an interface lets you organize things spatially in physical (or graphical) space, it can make you a lot more productive.
That said, you're right that physical paper is wonderful. I often find myself wishing for some way of moving easily between digital and paper representations of a problem I'm working on. The best I've come up with practically so far is heavy use of index cards and a scanner when I want to get things back into the digital realm. I think something like this could be slick and nice, but my current implementation is actually pretty clunky, so I tend only to do it during the initial phase of a new project -- print off all the info I think I'll need, grab a stack of index cards and a mechanical pencil, walk away from my workstation and sit down at the kitchen table to think and scribble, then scan in anything that needs scanning in and work the rest of the project digitally. I'd love to have the ability to move back and forth with less ceremony, but so far I haven't seen anything that would facilitate this.
I think the reMarkable (2?) tablet might be at least part of what I want, though I haven't had a chance to look into it very deeply. They probably also depend on a cloud service that could go up in smoke at any time, which I'd like to avoid for something super integral to my daily workflow.