Order is in the international system, not at the smaller levels. U.S. hegemony, or any hegemony for that matter, cannot do anything to quell uprisings or putsches, nor should hegemons care. Hegemons, according to the theory, are concerned with setting up an international regime that favors it. The U.S. has done this. Indeed, De Gaulle recognized U.S. global hegemony back in the 1960s. The world reserve currency is the dollar, the U.S. navy protects trade and ensures open trade routes, Treasury bills are the default collateral for large-scale loans and are a major form of world commerce, the UN is manipulated by the U.S, etc.
So Putin, while everything you mention about is correct, Hegemonic stability theory is that 1) Hegemons will set up an international system to their benefit, in essence removing the anarchy of the system and putting in a hierarchy.
2) While hegemons will gain in absolute terms from this system, other nations will gain more relative to their size and to the hegemon
I may point out though, that there were putshes and uprisings throughout history no matter what polarity it was in.