Between the two...could you really picture Queen Elizabeth attempting to seize total control of the state--much less accomplishing it? Or the monarchs of Norway, Sweden, Denmark, Spain, Belgium, the Netherlands, or Japan?
I can picture a President attempting to seize power in a Republican system. In fact I can point to several specific examples from the past few decades, successful or otherwise.
I'm definitely not saying constitutional monarchy is the perfect system, at all. I'm just saying that after spending most of my life with the assumption that monarchies were just a quaint anachronism left over from days gone by, a sort of political appendix...I've started to notice that they seem to have interesting properties and robustness that other systems might lack. It's possible that the monarchy serves a useful purpose after all (actually...much like the appendix).