"Your ammo might not know the difference...but the concentricity gauge will. "
No, it won't and assuming it to be so isn't proof. Loaded concentricity comes from straight necks and a good seater, no press can influence either. All a press can possibly do is push the round case into the round hole in the round die and pull it back out. Anyone sweating a potential half thou or five thou "misalignment" in a ram is going to have a stroke when he notices the sloppy fit of his cases in a shell holder. And he would fall over dead if he looked at how loosely a CoAx holds the dies and cases.
I have an RC, two little Lee "C" Reloader presses and a well worn 48 year old Lyman Spar-T which no one would have ever called a "precision" press. As an experiment, I once selected 20 highly prepped and weight selected LC match 30-06 cases and loaded/fired that box of cases on each press using the same load and shell holder and dies and applied the same level of care. Twenty samples of anything isn't highly statistically meaningful but there was no significant difference in the measured concentricity from the presses, run out ranged from zero to just under 3 thou and the average varriation between each batch and there wasn't enough difference to make a difference on either the gage or targets.
Meaning there's no magic in a press, the press used is the least significant element in hand loading for accuracy. That's why some people get great results with Redding's big turret press which can't possibily have identical alignment of every die hole and there has to be some slop in the turret or it couldn't turn. Bottom line. a good loader will be able to do good work with any press and a poor loader won't do good work with any press.
The CoAx is a good press and handles spent primers beautifully but I don't care for the ergonomics. My RC has good ergonomics but tosses spent primers all over the floor. Lee's Classic Cast is big, strong and precisely machined; it catches primers like the CoAx and has even better ergonomics than my RC. I have no childish animosity or loyalty for inanimate objects, IF I had to replace my RC next week it would be with a Classic Cast, it's the better press.