The real choice is not anywhere near as important in life as the perception of choice, and so it doesn't matter that the child doesn't have a real choice, what matters is that in giving that [choice], you increase compliance.
And then now, here comes the science: Is there anything that science can tell us to get [the child] to be a really better listener and controlled better by [the parent's] behavior? and the answer is wildly 'yes.'.