If you must do bad things, do them all at once. Then create a steady stream of better than expected events. People will get over the shock of the bad, and will remember the good.
By contrast if you try to hold back some of the bad, it will drip out bit by bit. Then people will get scared about where the real bottom is.
I've seen his tactics play our over and over again during my working lifetime.
My take is that the advice in those books is great advice for those seeking to gain and hold power in organizations. All else being equal, you should want the person running your organization to have mastered those ideas and be able to apply them productively for your organization.
That said, having an organization full of people who want to gain and hold power is a very, very bad thing. While you want the people at the top to do so, you don't want every Tom, Dick and Harry with a big ego doing so. So you are going to see an anti-pattern where a bunch of people who have no business trying to play Machiavellian games, attempt to do so. And part of the the reason why they should not do so is that they are not suited by temperament or competence.
We aren't going to see it, we already see it every day. The books are poison.
That's not to say that there may not be better approaches, in some aspects. In my mind I organise Getting to Yes and How to Win Friends as books that are complementary, yet approach the problem from a co-operative perspective.
But we haven't totally vanquished that part of ourselves that the Art of War and the Prince were written to deal with. Even if you banned these books, people would independently derive their principles.
For example who among us wants to believe that once we come to power we should distrust those who got us there, and look for our best allies among those who had opposed us? Yet again and again in actual organizations those who think they are owed will try to take advantage of the situation, while those who have reason to fear that they never can be trusted will overcompensate for that.
The Prince is relevant in any organization where a strict hierarchy needs to be maintained. Which is why it was relevant here to the question of how a CEO who needs to implement a painful decision (in this case big financial cuts including laying people off) should go about it.