The US posture in all conflict related matters is security and silence. Whether they believed public opinion was on their side or not(and it was), they would be engaging all the same. I'm not saying it's right, but that hardly proves some secret fear or this absurd notion they were acting against the will of the American people.
When the expanded drone strikes were leaked Americans overwhelmingly supported them and continued to up until Obama left.
Developing strong, pragmatic and principled national security and defense policies.
www.cnas.org
Maybe if Obama was a Manchurian foreign policy candidate and he turned into a dove after lying his way to a nomination and victory he could pull it off, but thats pretty silly hypothetical to ponder, and given how powerful the fear of terrorism has been, and how strong the moniker of being strong on terrorism was, it seems unlikely in my mind.
American people are shit, and as shitty people we lock in our politics into really shitty path dependencies. Not sure why this is a controversial thing to point out that I am getting quoted endlessly.