Thanks for the suggestions! I wish there was an easy solution but I don't think there is...
The arms are definitely tight against the tires, to the point that it's often hard to get the arms off the tire when unloading. I tend to run my tire pressures pretty high which is why there isn't much deformation. I always make sure to line up the "spool" between the tire tread, then yank the bottom of the tire outward before tightening the arm to get the maximum compression.
I'm pretty sure the gold bar knob is tight but I'll have to check. I never actually change the angle of the rack so I haven't looked at it in a long time. I've looked at the black bar though and don't see much (any) wiggle there at all.
When pushing and pulling on the bike it's pretty clear the remaining wiggle (after solving the steering movement) is mostly coming from flex in the arms which just happens to be the main area they chose to reinforce with their Xtreme duty rack by doubling them up with spacers in-between.
View attachment 1057556
I don't seem to be the only one having this issue either. Mine is wobbling somewhere between these two, less than the first but more than the second.