That logic makes no sense when considering Obama also armed the Muslim Brotherhood and fostered the Islamic takeovers of Libya and Egypt.
Obama allowed for the MB take-over of Egypt because the alternative was telling the army to massacre the people in Tahrir Square.
Allowing for the fall of Mubarek, elections, and trying to work with the MB (while maintaining relations with the army) was seen as the best way to manage the situation. Sure enough, the army coup’d the Brothers, putting Egypt right back to square one.
As for Libya, it provided a welcome distraction from all the US-backed regimes under threat at the time (especially Yemen and Bahrain) since Ghadaffi was deemed more expendable than the other two, and had a small, weak regime (even then it took 6 months for the rebels to win even with air support).
Al Qaeda is more strong and present in Syria than anywhere else in the world right now, and there is NO reason for the US to support the Islamists in Syria other than to weaken Iran and Hezbollah. No reason at all.