So long as there are enough civil service employees left behind, the government will more or less function. Elected officials come and go, but except at the most senior levels, the bureaucrats are there "forever." An elected official isn't required for a government employee to issue driver's licenses, marriage licenses, building permits, employee paychecks, get the roads plowed, etc., etc., etc.
At one law enforcement agency where I worked, there was a time that if the Rapture had happened, only one or two officers would have been able to report for work
Another place would have been completely empty