To say that neural networks are empirical risk minimisers is just to say that they find functions with globally optimal training loss (and, if they find functions with a loss close to the global optimum, then they are approximate empirical risk minimisers, etc).
I think SLT effectively assumes that neural networks are (close to being) empirical risk minimisers, via the assumption that they are trained by Bayesian induction.
To say that neural networks are empirical risk minimisers is just to say that they find functions with globally optimal training loss (and, if they find functions with a loss close to the global optimum, then they are approximate empirical risk minimisers, etc).
I think SLT effectively assumes that neural networks are (close to being) empirical risk minimisers, via the assumption that they are trained by Bayesian induction.