When banks in a country run into trouble or a bubble is forming, outside observers tend to attribute part of the problem to weaknesses in the national regulatory and supervisory regime. They naturally ask if other undetected problems may be brewing in the financial system in question.
This may lead to deposit withdrawals or to a national risk premium to develop even if the Sovereign has sufficient capacity to backstop the financial system. To my mind, a SSM, provided it is well-equipped and resourced, is less likely to have its credibility eroded in such a situation.