Why do we never learn cross validation in regular statistical textbooks? I read various stats/biostats textbooks, including Casella and Lehmann's book chapter on regression. Most of time, the textbook will report a p-value for significance of the parameters after regressing against some model. Then there is a model selection procedure followed afterwards.
However, those books will never touch upon cross validation (CV) or talk about using a test/training split. I learned CV and Monte Carlo cross-validation (MCCV) from machine learning books and rarely have I seen stat books covering CV.
Why were we not taught cross validation in stats? Or is it not practiced by statisticians in general? Or somehow that model selection procedure becomes superior to using testing data for model selection? Does a biostatistician/practicing statistician use CV in model selection in general?

I can't say with 100% certainty, but I can give you my two cents.
Let's start with the difference in philosophy between statistics (as practised in the books mentioned) and machine learning.  The former is (usually, but not always) concerned with some sort of inference.  There is usually a latent parameter, like the sample mean or the effect of a novel drug, which requires estimation as well as a statement on the precision of the estimate.  The latter (usually, but not always) eschews estimating anything except the conditional mean (be it a regression or a probability in the case of some classification models).
Thus "model selection" in each context means something slightly different by virtue of having different goals.  Cross validation is a means of selecting models by means of estimating their generalization error.  It is, therefore, primarily a tool for predictive modelling.  But statistics (again usually, but always) is not concerned with generalization error as much as it is concerned with parsimony (for example), hence statistics don't use CV to select  models.
Prediction from a statisticians point of view is not absent of cross validation.  Indeed, Frank Harrell's Regression Modelling Strategies mentions the technique, but that book is primarily concerned with the development of predictions models for use in the clinic.
