I would like to override a Javascript built-in function with a new version that calls the original (similarly to overriding a method on a class with a version that calls
I'm assuming your question is how do you overwrite a built-in and still be able to call it. First off as a disclaimer, you should never overwrite built ins unless you have a good reason for doing it since it will make it impossible to debug/test.
This is how you would do it:
window._alert = window.alert;
window.alert = function(str) {
if(console) console.log(str);
window._alert(str);
}