Training data attribution (TDA) is concerned with understanding model behavior in terms of the training data. This paper draws attention to the common setting where one has access only to the final trained model, and not the training algorithm or intermediate information from training. We reframe the problem in this "final-model-only" setting as one of measuring sensitivity of the model to training instances. To operationalize this reframing, we propose further training, with appropriate adjustment and averaging, as a gold standard method to measure sensitivity. We then unify existing gradient-based methods for TDA by showing that they all approximate the further training gold standard in different ways. We investigate empirically the quality of these gradient-based approximations to further training, for tabular, image, and text datasets and models. We find that the approximation quality of first-order methods is sometimes high but decays with the amount of further training. In contrast, the approximations given by influence function methods are more stable but surprisingly lower in quality.