That power is limited by the fact that the president is not really free to do anything he pleases from what's allowed, but is even before he hits the office tied by tons of deals, negotiations, behind the table alliances, special interests, donors, party loyalty, and, one would imagine, down to pure old blackmail and/or payola in some cases.
Nor are members of Congress 'free to do anything [they] please[]', but are also 'tied by tons of deals ...', making them, by your definition, also 'puppets'.
Nor are the Koch brothers, favorite targets of the anti-corporatists, free to do anything they please, but are also limited by many of the constraints you mentioned.
If everyone is a puppet, then the term 'puppet' is meaningless.
