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precious time would be wasted as that power is realigned
Serious question though: is the current situation not one in which the civil service left to its own devices + some caretaker PM might actually be more quietly efficient than all the politicians on top? Are they really adding all that much at the moment?
Sure, at least in the UK they're maybe not openly obstructionist like in the US, but still...
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The Civil Service is many things but it looks to government to set its priorities.
What legitimacy would a 'caretaker' PM have? Very little to none. Unless we have some kind of election or parliamentary vote to see who is the most 'popular' but then you are back where we are now, except you've wasted a whole bunch of political will and time to do that whilst the world looks on in bewilderment. And what Pandora's boxes would you open on the way.
And if you think Black Wednesday was bad you wouldn't have seen anything yet.
He is where the various factions focus their power. And he has (some) legitimacy with some of the electorate. Again, sorry if you do not like that, I do not like it much either. If he was to go, precious time would be wasted as that power is realigned. It is not something we need right now.
All that before what it might say for our country's ability to function in this new world we find ourselves in, should it turn out that our PM both gets it and gets incapacitated by it despite all the resources thrown at him to make sure he pulls through.