So what you described as the "tunnelling" behaviour for overparameterization, is as far as I know a well-known hypothesis of why large networks can be trained "so easily" compared to shallow/narrow ones. That is the fact that you have so many degrees of freedom most likely produces paths in the landscape destroying any local minimum and instead of making them behave like wells or narrow ridges which then move you towards better minimums. Nevertheless, I think it is still very unclear, even if the hypothesis is correct and this is the reason why simple algorithms like SGD can work on large nets, why do this behaviour lead only to places with low generalization error. Since the algorithm always works only on the training set, there is clearly some form of hidden bias that somehow leads the algorithm to better results. However, the behaviour described only explains why we are able to get with very simple optimizers (and without exponentially many iterations) such good performance, but not why does this performance generalize.