I'm really flattered. My golly we take our American values around the globe and we're shocked the rest of the world hates us. Like I give a damn. Our country's mission is to make sure another 9-11 is kept well and far away from our shores. If that means being a "benevolent hegemon," so be it. No one else is certainly capable of keeping the peace, making short work of our enemies, and keeping America secure for generations to come. America as always is still truly, "the last, best hope of mankind."
You seem to think that we are in a two option only scenario - either cower at the hands of terrorists or subjugate the whole world. Perhaps having armies stationed the world over, projecting force, smashing nations then rebuilding them, bribing other nations, financially aiding even more, peace keeping missions, nation building, etc. will keep America secure for generations. Then again it could bring blow back and overstretch - there is 5000 years of historic precedence for the latter, none for option one.
Since option one is a recipe for more wars not less and of course since war is the health of the state don't plan on having a smaller government or your taxes lowered. Are there other options? Yes very workable ones that would leave us peaceful and prosperous but they aren't too popular these days - peaceful trade with all, no entangling alliances and not striding the world seeking monsters to destroy. More war equals peace is rather Orwellian but then it fits with the zeitgeist.