I'd say the president, regardless who's currently in the WH, always has had this kind of power. While various laws, prohibitions (it started with Ford) and proclamations may state otherwise, he's got the wherewithal to go thumbs-down on just about anyone, seems to me.
Who's going to tell him, "No, Mr. President. That's considered assassination and you can't do that."?
Some things just don't need to be written down.