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?

Descriptive vs. predictive
In, say biology textbooks, the focus is on  describing the relevant data; the sample mean was…, the p-value for this result is… etc.
In machine learning texts, the focus is on producing models that can generalize beyond their training set; thus techniques like cross-validation are required in order to get a handle on that feature.
This oversimplifies a bit, but I think captures the essence of the difference in approach that some works take relative to others.  These differences are a reflection of the different priorities and goals for these different domains that apply statistics.
Consider the Theory of Point Estimation by Lehmann and Casella 1998.  They start chapter 1 with the statement "Statistics  is  concerned  with  the  collection  of  data  and  with  their  analysis  and interpretation.", which reads to me that these authors have chosen to focus more on applying statistics for analysis and interpretation rather than on applying statistics in a manner that can be reliably generalized to other cases.  Similarly, they describe the result of "classical inference and decision theory" as " Such a statement about [the estimated values of the model parameters] can be viewed as a summary of the information provided by the data and may be used as a guide to action." -- again the focus is more about how to describe the data (any maybe as a basis, with some interpretation, on how to guide actions), and less about the nature of the problems involved in ML.
Finally, let me emphasize: this is a matter of what particular authors, problem domains etc. have chosen to focus on in different areas.
