Dear John ,
4420 c.p.m not a bearing frequency of this fan motor ,But the input from the intermittence team is , there is a backlash between the gears in Gearbox .
Why are you so certain the 4420 cpm is not a bearing frequency. I see for the 6313 a BPFO of 3.07 which yields 4530 cpm. That is for 9 elements and often depending on the bearing manufacturer you will find +/- 1 element.
Backlash can be normal, the lack of it is abnormal in most instances. What are the tooth combinations, speeds, etc. If you have the long spacer coupling as you've shown you are also not likely to get gearbox vibration at the motor which is your concern unless I'm misunderstanding things.