Lee sent a thoughtful reply this morning, here's the key part:
"Try reinstalling the index rod in your press and then take your powder drop test weighs from the normal 4 stop cycle of the turret. The start stop action of the powder measure traveling with the rotating turret will agitate the powder in the powder hopper and metering chamber which will tend to settle and uniform the powder charges dispensed. If you incorporate all dies in the reloading cycle the additional vibration will help to enhance what should already be more uniform powder dispensing."
I reinstalled the index rod and did 20 powder throws. The results were just marginally better (50% of throws within .1 grain of target, vs. 40% the first time, and 65% within .2g vs. 60% the first time).
I think there two parts to the Lee recommendation. Rotating the turret, and using the other dies in series just before charging. The vibration from just advancing the turret all the way around between throws - which is all I did - is significantly less than that from actually doing the two actions prior to charging (resizing and decapping, priming).
I will do another test run where I prime and charge. Resizing - a fairly violent action with 30-06 on my turret press - is out of the question, as I tumble after resizing to remove lube. So I'll prime and charge, see if the pattern is any better, and report the results.
For those following, have you used the Auto Drum, with a stick powder like 4064 - either single-stage or indexing the turret and doing other reloading actions - and achieved highly consistent throws? I know the consistency can be very good with flake powders, from my handgun loading. It's stick powders that are at issue here.