That a government structure is designed to operate at the highest decision-making level does not mean that it should be beholden to its ministers. Wars are not won by generals alone, after all.
The excuse that a crisis is not the right time to replace ministers because their successors would have trouble being brought up to speed, does not hold water.
Such an argument would make sense only if a lot of valuable time and political capital had not been wasted already in covering up for certain cabinet members who quite clearly are not on the same page as the central government line.
In a crisis, hesitation can come at an even greater cost.