Your question is very reasonable from a western point of view. However, the Chinese think and organize their activities differently.
The law implementing the Work Permit system for foreigners began in May 1992. The Income Tax law predates that by about sixty years. As a result, the Tax authorities continue to limit their attention to whether you are paying tax on your income (or “paying tax on a reasonable level of income to support your livelihood in the ROC”). They do not really know about or care about the technicalities of whether you have a Work Permit or not.
In regard to the situation of an average foreigner, your statement that “you can only have one income in Taiwan” is, technically, correct, however it is not enforced in any systematic way by employers or the Tax authorities.
Consider this: Suppose that you went and gave a speech at a local university, and they paid you NT$5000, deducted 10% tax, and gave you the withholding statement. Is that illegal and going to get you kicked out of the country? I have never heard of such an event happening. (However, I have heard of people performing in nightclubs for one night who were picked up by the Police for working illegally.)
The one restriction that does exist is that as an average foreigner you cannot have two work permits simultaneously. To my knowledge, that is the only restriction that applies to the situation you have brought up.