If overfitting scores better on training data, is it not the less bad failure of the two?

Your intuition that a too-flexible model is easier to pull back is actually sound, and it is worth keeping rather than discarding.

The standard workflow really is to build something with enough capacity to overfit and then constrain it — regularisation, early stopping, more data, fewer features. Deliberately overshooting and reining in is a normal and effective way to work, because it separates the question "can this model represent the pattern at all" from "is it generalising".

The distinction is between overfitting as a stage and overfitting as an outcome. Passing through it on the way, with held-out data telling you when to stop, is method. Shipping it is the failure. Your reasoning was right about the process and wrong about the destination.

It breaks down on the phrase "does well somewhere". Training performance is not somewhere. It is the one place where the score is guaranteed not to mean anything, because that is the data the parameters were chosen to fit.

The asymmetry that matters is about how bad each failure can get:

• Underfitting has a floor. A model that is too simple converges on the average behaviour. It will be consistently mediocre and roughly as mediocre on new data as on old.
• Overfitting has no floor. A sufficiently flexible model fitted to noise can produce predictions that are arbitrarily wrong on new data — not slightly worse than the mean, but far worse than predicting the mean.

So they are not two symmetrical errors. One is bounded and the other is not, and unbounded downside is the thing you protect against first.

The other half is that they are not equally visible, and in practice that is what does the damage.

An underfitted model announces itself. Training error is bad, validation error is bad, everyone in the room can see the model is not good enough, and nobody ships it by accident.

An overfitted model looks superb right up until it meets real data. If your validation split is not clean — leakage, duplicated rows, a time series split randomly instead of chronologically — it can keep looking superb through evaluation too. The failure then surfaces after deployment, which is the most expensive place to discover anything.

A failure you can see is cheaper than a failure you cannot, even when the second one has a better number attached.

Small framing that helped me: ask what each failure would cost the person using the output.

An underfitted forecast is useless in a predictable direction, and people route around it. An overfitted one is confidently specific and occasionally absurd, and people act on it before they learn not to trust it.