Yes, NJ, I think we all know that. The reason this arose originally between Ludwig and myself was the fact that - a few years ago - Tories on AB and in the country/press at large constantly whined on about how Gordon Brown was (quote) "an unelected Prime Minister". You must remember.
When the relevant date arrived recently, I pointed out here that Cameron had now been an "unelected Prime Minister" for longer than Brown had been...ie his party had not received an overall majority from the electorate, yet he was still in No 10.
Having said how the system works, you must also be aware that there are multitudes of people who DO vote for the local representative of the party whose leader they WANT to be Prime Minister. Many of those would doubtless be put to it even to name or recognise their particular candidate in the relevant constituency! If they wanted Maggie and - the Lord knows why - they did, they voted for Jim Bloggs or Sylvia Stevens or Uncle Tom Cobley or whoever the Tory candidate was, knowing that their success would achieve what they wanted. It is, therefore, not totally true to say Prime Ministers are not elected, given that they often are "by proxy", as I've just explained.
I'll leave it there.