"Nothing to do" is a bit strong. It has something to do, and the explanation is in the history of how these games worked.
Can it be simulated on GPUs? Sure. But it was typical of software renderers of that era, and NOT typical of GPUs, and it has to do with how they were used.
"Software" isn't the point. You can do the exact same math on CPU or GPU. OpenGL supports nearest neighbor texture filtering on whatever hardware you run it on since 1992, one year before Doom even came out.
I isn't "simulated" by GPUs, it is a rendering mode that GPUs had since before they were called GPUs.
The reason software rendering look different from 3D accelerated games (incl. 3Dfx) is nearest neighbor filtering and low resolution, which were necessary to keep a reasonable framerate. At the time, it was considered ugly, so if your hardware supported bilinear filtering for free, it was turned on, but it didn't have to.
In the time of fixed pipelines (3Dfx to GeForce 3), you couldn't do arbitrary operations on pixels, so some effects, like the kind that are common in the demoscene were software only. So games that used these effects, like Comanche voxels can indeed be thought as representative of software rendering, and they can, of course, be simulated on modern GPUs as everything that was done on software before can.
But the article uses Quake as an example. I may be wrong, but I don't think it does anything that can't be done by a fixed pipeline GPU. In fact, even keeping the same style (nearest neighbor filtering + low resolution), hardware rendering will probably look better because the GPU does proper perspective correction on every pixel. The tricks used by the Quake engine are awesome, but they don't beat raw power.
This is incorrect. It is exactly the same algorithm as circuits in the GPU or as instructions on the CPU. No difference, except one: the way data is pipelined on the GPU makes texture blending free, or nearly so. So to show off, every 3D game back then turned it on (an OpenGL/Glide/D3D configuration option) and everything got smoothed out, often with no option to turn it off. That's why it ends up looking different: because the game itself selected different interpolation in accelerated vs software rendered. It was a game choice.
These days you can use different presets to override those choices, and have your 8K, 240Hz Quake look software rasterized, if you want.