I think that it's intellectually myopic to argue that this administration has the legitimate authority to pursue and kill those Al-Qaeda terrorists when the administration is at the same time arming Al-Qaeda terrorists.
I think you’re confusing legal authority and moral authority.
Under the 2001 Congressional Authorization to Use Force I think their legal authority is impeccably constitutional.
With regard to their moral authority you may very well be right.
However, it’s also not unreasonable to point out that we are (probably) supporting groups who are fighting Assad in Syria. He and his family’s regime have been undisputed enemies of America and supporters of terrorism for decades.
There are no groups in Syria right now that most Americans would consider to be “the good guys,” with possible exception of groups with little support from any actual Syrians.
So we have the option of just staying completely out of the mess there, or supporting groups that are more or less objectionable. The staying-out option, of course, leaving the territory fully open to Russian, Iranian or Chinese influence.
We allied ourselves with Stalin for 4 years to defeat Hitler, and we supported creeps like Somoza for decades to defeat Communism. It doesn’t seem unreasonable that we might similarly support groups we don’t like in the Middle East.