In fact, if it really worked it would not have accepted a drunken and senile Churchill for so long or allowed an unstable personality like Eden to take office. An American president is constrained, above all, by a written constitution and the separation of powers. The truth is that, lacking a formal set of restraints, a premier like Thatcher could and did play fast and loose with the system. The country is not at war, he's not ramming through Attlee-scale reforms and his majority is large enough for him to allow parliament a strong say and still know he'll win. At the end of the day, a great deal of this presidentialism is quite obviously smoke and mirrors, PR without content.
British premiers are usually frauds when they philosophise.
Attlee and Thatcher stand out as premiers because they inaugurated an age in their image and changed the terms of debate, but neither was a philosopher - Thatcher's pretensions in that regard were simply embarrassing. One is driven, ineluctably, to wonder about the sadly inverse nature of political leadership. When the American republic was young and the population still tiny, the Presidency produced Washington, Hamilton, Jackson, Jefferson and Lincoln. Two hundred years and two hundred million people further on, with everyone far better educated, richer and more sophisticated, it could only produce the likes of Nixon and Ford, Reagan and Clinton. Later, still with a poorly educated population under thirty million, it produced Palmerston, Peel, Disraeli and Gladstone.