Who gets to determine what a "legitimate" target is? How do we know whether the administrations isn't just trumping up charges against innocent people in order to spy on or summarily execute political opponents? With no checks on executive branch to oversee its actions, the potential for abuse is very high. (Unless you think that the government always does the right thing and would never abuse its power.)
To allow the executive branch the power to do whatever it wants without ANY checks is to essentially make the executive a king. If the government can accuse an innocent person of a crime and take away a person's rights and due process based upon the executive branch's say so alone, then the government can do that to everyone and the protections promised in the Constitution are meaningless.
When has there ever been due process or checks and balances in time of war?
AlpQuaeda is at war with the USA. That is just a fact. We can ignore them or go to war with them. But if we go to war, by definition there won’t be anything resembling due process.
I understand your concerns and this is indeed a slippery slope. But when it is necessary to cross such a slope to defend American lives, you do so, and protest when excesses are proven, not because excesses CAN happen. If that tack is taken, there can be no effective self-defense.