Lies have proved very powerful against the truth. People give power to Satan by believing lies.
That doesn't explain why Satan is given power over the ruling authorities, why he's the "prince of the air", and why he has the power over those who aren't in the Church. What does believing lies have anything to do with it? The only way that's accurate is by saying that those who believe in the total Truth are protected from him, which is what the NT implies, but everyone else (i.e. the 99.999999% rest of the world) is under his power.
God gives bodies to faithful angels to accomplish God's will. OK. I can believe that! Does God give bodies to unfaithful spirits to accomplish havoc? Why?
Why not? Why would he give power to Satan to rule the authorities of the Earth and be the "Prince of the air" and the "god of this age"? I think your understanding of Satan's power is not congruent to what the NT says.
If the answer is NO! then where did the unfaithful spirits get their bodies? Let's talk Halloween. It's near, you know. Did they use dead bodies? It's getting worse and worse for those poor ladies of Noah's day.
Well the answer is not No, the answer is a default yes until you can prove that he deliberately took this power from them. Until then, you have not disproven the NT claims that Satan is in control of pretty much the whole world and your answer of "people believing in lies" does not address this whatsoever except "Those who don't believe in lies are under God's protection from him", so if Satan is given that power over disbelievers, why not let the fallen Angels have power as well?