If they succeed a man or woman who has not been elected by the public would replace a man who has himself not been elected by the public.
This is without precedent - in this country at least.
Not true, since Neville Chamberlain took office in 1937, some time after the 1935 general election, before being replaced by Winston Churchill in May 1940. He is the only 20th century Prime Minister never to fight, win or lose a General Election as a party leader.
And, if you wanted to be super-super-pedantic (oh, go on then!), even if you added "without precendent in peacetime", that still wouldn't quite be right either.
Not a lot of people know that the Conservative government resigned, and let the Liberals govern, in December 1905 before their landslide defeat in the January 1906 General Election.
Which meant that Arthur Balfour was selected, without a General Election, in 1902 to replace his Uncle Lord Salisbury as PM, and that Sir Henry Campbell-Bannerman replaced him as Prime Minister in 1905 before the two men fought a General Election against each other in 1906.
Despite astonishingly using the Tory campaign slogan "We can't go on like this" today, I don't think Patricia Hewitt or Geoff Hoon are proposing that the Labour government should follow that precedent.