You are speaking in very broad terms here when you mention prestige. Also the division you do at the end is not completely correct.
No one is saying that Yale is way less prestigious than Stanford, of course not. However the fact remains that Stanford is super strong in all fields across the board while Yale is definitely relatively lacking in some STEM areas. For example, an engineering degree from Stanford and an engineering degree from Yale are worlds apart in terms of perception and prestige. A person interested in tech would prob not even consider turning down Stanford for Yale.
On the other hand, Stanford is a very viable option for Finance, Wall Street and Law school given its great strength in these areas. (Stanford has a better Econ department and much better business school than Yale, while its Law school is right behind that of Yale, and also Stanford is ranked the same or higher in most other humanities, social science fields). Also one could argue that there are higher chances of braking into Wall Street from Stanford since not as many students view than as the ideal job as at Yale --less competition.
Also no one ever says that Stanford is not good enough to be Ivy league. That would be laughable since Stanford has surpassed all of the Ivies in many metrics, including Harvard, let alone Yale.
No one is pushing an agenda. These are just the hard facts.