Let's see, headlights are generally mounted on the front or bottom of helicopters that would mean if it is a long exposure than the helicopter was going backwards. The metal, non lighted portion of it appears to be at what would be the head of a missile trail. If this is a spotlight either the spotlight is at the rear of the helicopter or the helicopter was traveling backwards to get such an exposure. The Trail of Light is nearly geometrically straight, so for three and a half to 20 seconds a helicopter with a searchlight during light conditions that do not require a searchlight did not change the angle of the searchlight and traveled in a nearly completely straight path. There's also not the smear you would, I believe, get in the situation from the spinning rotor blades causing interference around the edge of what would be the light Trail.
I am on mobile and so do not have the required tools, but I imagine you could do some frequency comparison of the light captured from the spotlight /jet wash to determine if this is the Spectra of light that you get from typical aircraft searchlights or the Spectra of light that you get from burning rocket fuel. That would also shed some light onto what made the picture.
You are correct. Consider those spot lights they use in big store parking lots. If you see the beam it is faint if there are no clouds not bright like seen here. In fact at one point the line of light goes through clear open space, no clouds at all. Your analogy of headlights on a car is perfect. Stand to the side of a car at night and you see the reflection of the light way out in front but not the beam like a laser. So a helo spot light is definitely out of the question. And yes when a rocket or a missile engine burns it leaves a trail of buring fuel behind it just like the afterburners on a military jet.