... once you have a chief executive chosen by popular election, you are in trouble. The “extreme” measure of the legislature removing the head of state happens much more readily in parliamentary systems ... we are just about the only country without a parliamentary system that isn’t pretty far along on the autocracy spectrum.
... the problem with popular election of Prime Ministers, and especially of Presidents, is that they become much more powerful than legislatures. They have national legitimacy, they can present a unified front, and they can dominate the news media. Separation of powers is a pipedream.