You could certainly break up training data, but breaking up the models into mini models based on which training data is used wouldn't work with neural networks trained using gradient descent. Basically whatever the state of the model is it depends on the totality of the training data that it has been trained on (and the order) and it isn't possible to go and remove the effect of a specific training data point without then retraining for all of the data that followed that data point (and even that assumes you were storing a snapshot of the model before every single training data point, which I doubt anyone does)

However, that's no excuse and it is of course possible to entirely retrain a network using a clean dataset and that is what these companies should do

Yes, but that's not easy... I can't remember exactly, but I think I saw an estimate that the compute time to train just one of the GPT models cost around $66 million. IDK whether that's total cost from scratch, or incremental cost to arrive at that model starting from an earlier model that was already built, but I do know that GPT is still to this day using that September 2021 cutoff which to me kind of implies that they're building progressively on top of already-assembled models and datasets (which makes sense, because to start from scratch without needing to would be insane).

You could, technically, start from scratch and spend 2 more years and however many million dollars retraining a new model that doesn't have the private data you're trying to excise, but I think the point the article is making is that that's a pretty difficult approach and it seems right now like that's the only way.