Good theories.
I know most of the time there is sexual assault involved in these types of killings (young girls).
But I circle back to two theories. One is that the thought of killing itself is what grips him. Not of sexual assault. But of being able to control a person and take the light from their eyes. In some weirdos they just have a strong urge to murder. It can give them a sexual excitement or just satisfy a blood lust.
The second (which matched more to me with the first sketch or with an older guy) is the theory of a disgruntled, martinet. The kind like George Zimmerman and others we've seen who fancy themselves as authority figures and want to discipline others but who lack what it takes to actually be in law enforcement or successfully in the military. They have feelings of entitlement about the status they should have and a deep desire to be a respected authority figure, but their psychological make up precludes them from being accepted to any such position because they're weirdos and it becomes apparent to those who are capable of such positions.
If they were able to get such jobs they'd likely use those positions of authority to brutalize others, without reason.
They desperately want to use a weapon against someone they view as needing to be punished. "Your grass is a couple centimeters too high. I'm going to fine you." "You don't belong in this area. You must be up to no good. I'm going to detain you." "You aren't allowed to be on this bridge. You need to come with me."
Those are the theories I'm leaning toward.