The idea goes back to England’s Glorious Revolution, where MPs fought hard to put the Crown on a short financial leash, so that they could control Crown officials’ actions. Although they did not use the term, English arguments about what would give Parliament bargaining leverage vis-à-vis the Crown hinged on the budgetary reversion. Because expenditure authority would lapse every year, forcing portions of the government to “shut down” in contemporary American parlance, parliamentarians were assured the Crown would seek a new budget every year — whereupon they could bargain for attainment of their various goals.
Eventually the power of the purse meant the monarch became a figurehead, the House of Lords a rubber stamp and power truly resides in the House of Commons. So, the analogy would be that the President becomes an (elected) figurehead, the House of Lords becomes an (elected) rubber stamp and the House of Representatives becomes the center of power. And the Tea Party Republicans want to turn us into Britain!