You are onto something. And the legislature gets their power from where? Look at any state constitution or at the U.S. constitution(s)--who gave the legislature power in the first place? Where does any Commander In Chief get his/her military imperium from?
As for your comment regarding fictions or franchises you seem to be touching upon what is called 'plenary power'. But even still, where does the legislature get its existence and power from? Did you know that the Vice President of the United States is the president of the U.S. Senate (i.e. presides over the legislative body) and is thusly part of the legislature and is actually referred to as "Mr. President". Veto power of the President should make more sense in that light--because in a sense the Vice President is just the underling of the President and so that ties the president to the Senate (i.e. President of the United States).
In the Constitutional sense, a president presides sessions of a legislature. The president as an executive officer over the executive branch sense is another matter (perhaps Chief Inspector General could be an appropriate title). The executive functions extend from the legislation, subject to check by the judicial.