Update: after the fix the counter seems OK again. 12 ps standard deviation (61k PPS-pulses, collected over 17 hours) on a cable-delay test:
A broken CH1 input trace seems to be a common problem on the Agilent/Keysight 53230A counter. This is the second unit with the same problem we have seen.
The fix is to bypass the broken trace with a wire directly from the center of the input-BNC to the next unpopulated SMD-pad.