Wouldn't a tribe that is able to do as I can have the greatest evolutionary advantage and thus outcompete the tribes requiring specialization like you describe?
I'm sympathetic to your situation -- but with all due respect, this would seem to be a strategy with a vanishingly small likelihood of benefit.
"Trying" to sleep is also doomed to failure, of course.
Personally, what works best for me is modeling solutions to issues in projects (work or otherwise).
I originally tried this, hoping for the benefit of "sleeping on a problem". I don't think it's been particularly successful for that (though it could be subtly so, I guess), but it has about a 90% success rate for inducing sleep within 15 minutes. :)
My solution and yours might be the same thing (distraction from the goal of sleep), but I'd worry about the stimulation of video, the light from the screen, disturbing partners, etc. And also that I might get interested, and stay up for hours instead (this is my failure mode for reading before sleep).
Thinking about projects prevents me from sleeping. It's the surest way to keep me up with my mind racing for a few more hours. If it works for you, great!
The "mode" for watching Netflix for 15 minutes is turn it on, watch for 15 minutes, turn it off. I won't claim it works for everyone, but it does help settle my mind at night.