I have to work the problem backwards.
The burglars were employed by the CIA.
The fact that they were the CIA is all one needs to know.
The CIA committed a crime for which the President took the fall.
Why did the CIA do it; doesn't matter.
What were they after, doesn't matter.
What matters is the CIA caused the President to be replaced.
That makes it a coup (planned or unplanned).
As Dr. Phil says, "you don't reward bad behavior."
Yet in that instance you can see who retained the power - the CIA.
If we assume the President said "find someone to break-in",
the hiring agent intentionally or unintentionally, hired the CIA.
But again, did the President make the order? And if so, does it matter?
See also:
Dear Jim: Re: Nixon and JFK