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?

The main reason is that almost all book authors are in inference statistics. In particular, bio statistics is heavy on this aspect. A lot of stats used in regulated industries, such as banking is guilty of this too. Question like "what caused this? Did this cause that?" are usually asked from the inference point of view.
Cross-validation is of interest to forecasters. If you run into a book written by people who are in predictive field, then you should see the discussion of cross-validation. A good example is Hyndman's book, see here. I personally look at p-values mostly when asked to, in our industry we have governance folks who love this type of nonsense and request us to show a ton of pointless metrics.
