Wonderful write-up of attempting to tackle this problem. I believe there must be a significant number of people who have played both Minecraft and Super Mario Galaxy, and had something like this sequence of thoughts - although you have followed it all the way to an actual demonstration, and written up your thoughts along the way so clearly.

The vertical distortion is the biggest issue IMO, there are a few reasonably satisfying ways to approach the horizontal tiling of each “shell”. For example, you can make your world a donut instead of a sphere, and now you have a perfect grid at each level! Of course, this introduces a level of distortion between the interior and exterior, so you also twist the donut once, and now you’ve both solved your distortion problem and invented the stellarator fusion device.