Guys, thanks for all the troubleshooting tips and general advice.
There are two of these machines that were built to identical prints by two different OEMs (long story) and because there have been no technically-capable people touching them for years I concluded that the problem could not be a configuration problem.
I stepped through all 80-something parameters on both AC drives on the keypad and looked up the meaning of each one.
And sure enough, somebody had been messing with the one that didn't work, and it might have been back in 2012. They customized the V/Hz ratio and put in "custom boost" values that evidently were just enough to get the big carousel moving under ideal conditions.
When just a little more static friction developed, those boost levels wouldn't let the motor move at all.
I changed all the parameters to match the parameters in the drive that was running correctly, and what do you know... lots more starting torque !
These machines move very slowly; the "slow" speed was 2 Hz and the "fast" speed was 5 Hz. Maybe next time they'll put in an appropriate gearbox.
Again, thanks for all the troubleshooting pointers. Sometimes the most important part of troubleshooting is eliminating the things that aren't wrong.