AHC: Prime Minister Bertrand Russell

Is it possible to have Bertie become the Prime Minister of Great Britain, presumably under the Labour Party? As I understand it he stood for election in 1922 and 1923, but both times it was in a conservative safe seat and didn't expect to win. He would remain politically active for the rest of his life, as a public intellectual, supporting gay rights, nuclear disarmament, anti-imperialism, and various liberal socialist causes, but never sought political office. If he decided that the best way for him to bring about the changes he wants to see is to go into formal politics, how plausible is it that he could become leader of the party and win the election? Furthermore, what would his political career look like? Finally, what sort of political legacy would his administration leave and how would he be viewed by contemporaries and later generations?
 
Compared to someone like Orwell, I've always found Russell's later(*) political writing(such as it is) to be rather cursory, and somewhat dillentantish. He seemed to focus on a few issues that were of ethical concern to him, broadly endorsed Anglo-American liberal values, but other than that, didn't pay much attention to the everyday nuts and bolts of partisan and party politics.

So, I don't really get the impression that he'd have much of a stomach for the long climb up the greasy poll.

(*) I say "later political writing" because I know he wrote stuff on German Social Democracy and Russian Bolshvism earlir on in his career, but I don't see much evidence of such interests in his canon after that, save that it's clear he maintained his dislike of the Soviets.
 
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