Of the famous founders, over half were Stanford undergrads and therefore likely were "coterm" students. That means they just added a year to their degree and got this degree tacked on. That saves lots of time and money compared to going to Stanford as a master's student. There are a lot of things that are "worth it" if you don't have to move apartments/cities and get it for half the price — but which are not nearly as worth it if you're paying double and add the friction of moving to the area in order to enroll.
I’m not sure how it factors into the overall admission statistics but getting accepted for a coterm is, or at least was, significantly easier and more straightforward. In my time it just meant a GPA above a certain cutoff, a letter of rec from a professor, and non embarrassing GRE scores. A very good letter of recommendation could make up for deficiencies in the other two. It’s not exactly a super selective elite club like the article implies if you’re already there for undergrad.
Can Stanford undergrads coterm in MS&E with any undergraduate major?
Yes, I knew people who cotermed in MS&E (and other masters programs) with a different undergrad major. I think you just need to make sure you fulfill whatever prereq courses they ask for but my knowledge is old at this point. I imagine you’d be fine going from any engineering major to MS&E, but if you were an English major who happened to take a bunch of math and physics that would probably work too.