Author here. This was definitely something I struggled with when creating this simulation. Reasonably fast simulations of tectonics and erosion have already been done before: some parts were tricky to translate to the constraints of a shader, but the parallelism pays off in being able to run the simulation at interactive speeds. The difficult part was the global climate model. I spent a while researching existing methods, but as you say, many of these are computationally expensive scientific simulations.
In the end I just downloaded some data for the Earth and went about seeing how well I could replicate climate maps with some simple curve fitting in a Jupyter notebook. After some trial and error, I was surprised to find that you could actually approximate things quite well with sine curves and some blurring, as described in the article.
Overall I learnt a lot about Earth systems from this project, it was a fun hobby for a few months. I have some ideas for taking it further, but haven't gotten around to it yet. Feel free to reach out to me if you have any questions, or want to discuss any ideas you have about it.