I did a test of three different firmwares and measured the current consumption. I used the same meter, same hardware, the only change was the Arduino sketch. I don't think giving my exact mA numbers would mean anything given a 10 to 15 % tolerance on components and meters, I will state that the Oscillator sketch measured at about half of the current used by the eitherkit sketch. That difference was 22 mA which to me seems like a lot. My personally developed beacon sketch was a 4 mA more than the Oscillator sketch, but that is apples/oranges because the beacon is changing frequency compared to the constant frequency of the other two sketches.
With this simple test I conclude that the firmware logic can have a significant impact on the power consumed by the SI5351A, and the 5v to 3.3v level shifting circuitry. I did not attempt to measure current flow in those two components.
An interesting, but understandable, measurement was that the base Arduino, without any additional hardware consumed the same current with all three firmwares.
I don't know that these results explains the root cause, but I think it confirms your observations.