I just find the blacked out background flash creates to be totally un-natural looking. If you do use flash, I'd use it at fill light levels not illumination levels.
My feeder is set up where I can adjust my camera position to the give me best results based on the angle of the sun. And I only grout when the sun position is right.For all my nature images I prefer natural light. To my mind flash is a poor substitute for working out an acceptable camera position. Flash to me just looks artificial.
I completely agree that 1/2000 is good enough for hummers, but you can get away with 1/1000 with minimal motion blur. Going to 1/6000s cost you 1.5 stops of ISO, and probably considerable loss of detail.
K-3 at 800 ISO.1/1250s. ƒ5. Using the limits of ISO (noise is getting bad) shutter speed, (starting to see some motion blur in the wings, a depth of field at ƒ5.) If you aren't pushing the limits with hummers, you probably aren't getting a decent image.