Surely that doesn't apply when both metrics are bad? I don't know. I'm not familiar with job reporting statistics. So info / perspective would be appreciated.
Also, when you say "over fits" I'm used to that in a machine learning context where it means data outside the fit is worse. Did you mean that ADP is more accurate with private sector and BLS is more accurate with public sector. Or is it more a matter of ADP comes in higher with private sector and BLS comes in higher with public sector. Usually, "overfit" is a bad thing in a machine learning context, but I wouldn't expect ADP to do worse in private and BLS to do worse in public. But the opposite. I would appreciate it if you could clarify the terminology from your perspective.
People that know account for these biases. This is why the ADP > BLS heuristic exists. If private sector jobs growth is less than government job growth, reflecting the biases in their respective models, that is interpreted as either the government sandbagging the numbers with make-work jobs or private sector employment being so poor that the monotonic government employment eclipses private sector job growth. Either way, it is a bad sign.
As a rough heuristic, ADP overfits to private sector jobs and BLS overfits to government jobs. There is a popular derivative heuristic that the economy is "good" when ADP > BLS.