AP5002 step-down switching regulator loop stability
Wow, thanks @Bruce for reminding me of the ground loop! I looked at the ground plane layout again, and was not pleased... I guess after clearing the ground plane under the switching node, I forgot to check the important should-be-low-impedance loops around the PSU.

Indeed, the input and output side grounds are far-far away from each other, and might be stitched with too few vias as well. I soldered a beefy cable between the input filter caps GND and the D16 diode anode:



And without further changes, the PSU is stable. Output voltage @ around 800mA:



With an additional 100nF output filter cap, the peaks can be dampened a bit:



Lesson well learned, keep your loops tight when designing a switcher regulator!