This demand has two separate but linked parts.
First, that our Prime Minister should be elected by the people thus preventing the party anointments that allowed Major, Brown and May to become PM. So at the time of elections as well as voting for a party you also get to vote for the PM you want.
Second, that the cabinet (executive), as in America, should be completely separate from parliament (legislature) thus allowing the whole of the House of Commons to hold the executive to account.
Returning to the first part let’s consider the legitimacy of David Cameron when he became PM. He gained office by virtue of 33,973 votes in the 2010 election. All these came form his constituency of Witney, which boasted 78.220 electors. The rest of the nation was not allowed a vote for the man. He may have been elected as an MP, but he was not elected as prime minister through a general franchise.
As to Parliament it should and must hold the executive to account but with our cabinet being mainly selected from the H of C as with ALL ministers, that totals around 200 MPs who are compromised, and if they know what is good for their careers will not want to rock the boat within their own party.
So when formulating this demand we concluded that ministers and other office holders cannot be members of parliament. If members become ministers, they must resign as MPs. As a consequence PMs must appoint their own ministers - from whatever source they choose - subject to parliamentary confirmation and dismissal. This has the added advantage of widening the recruitment pool.
As to our monarch she or he remains head of state with their duties unchanged. The PM keeps that title and while elected they are still PM and not a President.
Finally we would suggest the term of office for the newly elected PM should be, as in America, a maximum of eight years. However that detail shouldn’t distract from the basic principle that we should have an elected PM and their ministers must be separate from Parliament and held to account by it.