I had a similar hypothesis, so I built something like what you describe:
https://github.com/wpietri/sunrise
Currently I'm at only 4000 lumens or so in my main space, and I'd like it brighter. But my sleep is definitely better. I attribute it not so much to the daytime light, but to the way my apartment gradually gets dimmer and redder in the evenings.
I had set out to make something like a SAD lamp for my whole apartment. But as a side effect, I also basically made f.lux for my apartment, and I really like it. When evening comes, I generally have to fight to stay awake rather than fighting to go to sleep.