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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.
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...