And then some - hadn't really thought about it until your comment, but somehow we regressed from infinite orientations (well, in that plane) to one, and are now all excited about having two...

Are there any example of barrel-style connectors with >2 pins, i.e. coaxial barrels? Similar is the 'jack' (3.5mm and such) but with split regions along a single pin rather than concentric rings of course. I wonder why such designs aren't more popular/explored (that I'm aware of anyway) for multi-pin data connections?

There were barrels with three..

eg lenovo, two charger generations ago:

https://i.imgur.com/Wn4EM33.png

But usually the inner one was only connected via a resistor, to tell the device the wattage of the charger (but could be used by a bunch of 1 wire protocols instead.