Since you changed out the wheel and all is well, I would say there is play in the hub bearings in the removed wheel.
I get the exact same issue with my Sparrow; headset is good - perhaps just a tad too tight, but I like 'em like that - but there is a little bit of play with the front Phil Wood hub. Lock the front brake and rock: no issues. Pick up the front and lightly drop: rattle. Hub bearings are more than likely your culprit, too.
I'm not at all suprised at your assessment of the climbing ability of that rig - my Compact was the stiffest, best-climbing bike I ever owned.
DD