So, let me try to clear some of that ground. We do not have a directly elected Executive in this country. Maybe we should but that is another story altogether. Once out of the EU or standing beside its debris we might change our constitution and create a system that would, indeed, elect our government directly rather than indirectly through Parliament. So far I have not seen a single proposal that would create anything but a Premiership that was even less accountable than it is now.
Let me go through it all again: in the General Election we vote for individual MPs on a first past the post system and the party that collects the largest number of MPs, should it find itself in a position of being able to form a government, does so. Who becomes Prime Minister in those circumstances is up to the party in question to decide. Whoever is the leader of that party is summoned by Her Majesty and asked to form the next government. Presumably, she also adds sotto voce "and God help you".
This does not run smoothly all the time and there have been various occasions when the party in question, being in an overall minority in the House of Commons, has to make choices: its leader can form a minority government with the certain knowledge that there will be another election within a few months; he (the one time we had a she the question did not arise) can form a minority government with the full support of one of the smaller parties, usually the third one; or he can team up with another party to form a coalition, a tricky but not particularly unusual course of action in modern British politics.
As a matter of fact, there was no talk of any coalition during the 2010 election and it was formed against the wishes of elected representatives of both parties as well as those of the electorate. So, if any government is unelected, it is this one.
The argument about Gordon Brown runs as follows: when it is pointed out that his party happens to have won unequivocally three elections, with two of them as landslides, the response is that he was not the leader at the time so people did not vote for him. (They did, as it happens, in his own constituency but let that pass.) This is true and, perhaps, the Labour Party would have done better to hold some sort of a leadership election when Tony Blair announced his resignation. However, I repeat, that is up to them.
A new Prime Minister taking over in the middle of a parliamentary term is a very common occurrence. In 1935 Stanley Baldwin replaced Ramsay Macdonald, called a general election, which gave the National Government another huge victory. Baldwin was replaced by Neville Chamberlain in 1937. An election was going to be called in 1940, which the Conservatives would have won, but events intervened. Instead, another "unelected" Prime Minister came to power, Winston Churchill. No election till 1945, which was called with great reluctance on his part and which resulted in a Labour landslide. Then there was another Labour victory and in 1951 a grudging Conservative one.
Chuchill hung on till 1955 when he was succeeded by the "unelected" Anthony Eden, who called an election and increased the Conservative majority. The Suez debacle led to his resignation and he was succeeded by the "unelected" Harold Macmillan who resigned in 1963 to be succeeded by the "unelected" Sir Alec Douglas-Home.
Moving right along, we have an "unelected" Prime Minister in James Callaghan in 1976 and in John Major in
Let me also add that until 1965 the Conservatives did not elect their leaders, who simply emerged after a great deal of negotiation behind the scenes. Labour did elect but there was just as much negotiation behind those scenes before the vote went through.
Apart from Eden, all the post-war Prime Ministers who took over between elections, continued in place to the last possible minute after which the party they led was either returned with a smaller majority as in the case of Macmillan and Major or lost as in the case of Douglas-Home and Callaghan. No particular conclusions can be drawn from that.
I trust I have cleared some ground.