There used to be rules for the hand you write with, too. Left-handed writing was forbidden, and didn’t work well, anyways when using a dip pen that wasn’t specially designed for left-handed use.
I think the Japanese obsession with stroke order has its roots in writing with brushes and slower-drying ink. If you use them, stroke order will affect what letters look like more than when using ballpoint pens.
Not really. Even with ball point, while for neat letter stroke order doesn't matter, for a messier (normal day-to-day) writing, stroke order absolutely matter. It can become illegible fast if you use wrong stroke order, because correct stroke order tend to create similar overall shape while the wrong order don't. Granted, you don't have to be exact with the stroke order, but general rule still need to be followed -- you can't just write it in reverse order and expect it to be easily readable.
Quite a lot of Japanese people seem to recognise characters less by their shape and more by the stroke connections, in my experience. No matter what the tool used to write it, legibility really requires writing the characters in the correct order for the most part.
I think the Japanese obsession with stroke order has its roots in writing with brushes and slower-drying ink. If you use them, stroke order will affect what letters look like more than when using ballpoint pens.