Even though I called myself a statistician for decades, I am not sure what it means now. It has specialised into sub-fields: econometrics, financial modelling, biostatistics, machine learning, business analytics. This might be the sign of a matured discipline where the value has been inserted into many fields.
However, I think the material many students might miss is the general understanding of how to model uncertainty: concepts like explaining variance, bias-variance trade-off, experimental design, frequentist versus Bayesian paradigms, the difference between an estimate and an estimator.