That seems to rely on the school of thought that "there's no such thing as bad publicity". Perhaps you're right, and someone could be skilful enough as president to create crises, despite their limited power, and also trick people into thinking that the crises are someone else's fault, and still somehow able to take the credit when they solve these crises.

On the other hand, I think such a belief requires one to hold quite a low view of the public and democratic processes generally. You might say that experience supports such a low view, but I would counter that it applies equally, or more so, to the status quo, where all power is held by the prime minister, who has every incentive (and much greater ability) to create these crises (especially just before elections) and tell the country that they have to unite behind the government to survive.

The fact that such a strategy is already possible under the current system just confirms to me that there needs to be different roles, held by different people, with different incentives, to stop these extreme edge cases. I think that what is self-evident is that concentration of power leads to abuse, while diffusion of power leads to moderation and representativeness.