Your story about elected PM's is all wrong. The British public have never voted for a PM as that's not how it works.
The public vote for their desired MP and the leader of the party with most MP's becomes PM. If they resign or get kicked out the same party then votes for a new leader and therefor new pm.
https://uk.news.yahoo.com/what-happened-to-the-other-nine-prime-minister-elected-by-their-party-and-not-the-public-073618818.html
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