Comparing model selection methods

The standard textbook analysis of different model selection methods, like cross-validation or validation sample, focus on their ability to estimate in-sample, conditional or expected test error. However, the other interesting question is to compare them by their ability to select the true model.
To test this I have thought to generate data following the process y = x1 + ε and test two linear models. In the first one only intercept and parameter for x1 are estimated and in the second a random noise variable x2 is added.

I decided to compare 5-fold Cross-Validation, Adjusted R-squared and Validation Sample (with 70/30 split) methods by testing their ability to select the true model - i.e. the one without x2 variable. I run the test assuming initial sample size equal to 10, 100, 1 000 and 10 000. Here goes my code (corrected after Andrej-Nikolai Spiess comment):

library(boot)

set.seed(1)

decision <- function(n) {

      x1 <- runif(n)

      x2 <- runif(n)

      y <- x1 + rnorm(n)

      train.size <- round(0.7 * n)

      data.set <- data.frame(x1, x2, y)

      train.data <- data.set[1 : train.size,]

      validation.data <- data.set[(train.size + 1) : n,]

      formulas <- list(y ~ x1, y ~ x1 + x2)

      cv <- arsq <- valid <- list()

      for (i in 1:2) {

            cv[[i]] <- cv.glm(data.set, glm(formulas[[i]]),
                              K = 5)$delta[1]

            arsq[[i]] <- summary(lm(formulas[[i]]))$adj.r.squared

            valid.lm <- lm(formulas[[i]], data = train.data)

            valid[[i]] <- mean((predict(valid.lm, validation.data)

                           - validation.data$y)^2)

      }

      return(c(cv[[1]] < cv[[2]],

               arsq[[1]] > arsq[[2]],

               valid[[1]] < valid[[2]]))

}

correct <- function(n) {

      rowMeans(replicate(2000, decision(n)))

}

n <- c(10, 100, 1000, 10000)

results <- sapply(n, correct)

rownames(results) <- c("CV", "adjR2", "VALID")

colnames(results) <- n

print(results)

The results are given below and are quite interesting:

           10    100   1000  10000

CV     0.7260 0.6430 0.6585 0.6405

adjR2  0.6570 0.6735 0.6575 0.6965

VALID  0.6195 0.6275 0.6220 0.6240

Cross-Validation performance is the best for small samples but deteriorates with sample size. Validation Sample approach performance does not change with sample size and is a bit worse than Cross-Validation. Adjusted R-squared method is actually better than other methods for large samples (again - thanks for Andrej-Nikolai Spiess comment).

What is interesting for me is that increasing sample size does not lead to sure selection of the true model for Cross-Validation and Validation Sample methods. And sure - for large samples the estimate of the parameter for variale x2 is near 0 so the mistake is not very significant, but I did not expect such results.