> You can't start a side business on a regular work visa.

This is incorrect.

A friend of mine started a businesses on a work visa and I myself have even started a business on a student visa [1]. No legal problems. I just recently switched from "Student" to "Manager" visa, and immigration had no problems with the fact that the business was already running for over a year before I applied for my residence status to be changed.

[1] http://marui-plugin.blogspot.jp/2016/11/starting-company-as-...