I've worked a lot with Oxbridge and Ivy League folks and there is nothing particularly special about them. An Oxbridge degrees bestows an out-of-the-box premium personal brand, as you've demonstrated, as well as the social network, but not superior ability, in my experience
When I graduated, ahem, a few decades ago, the main difference between Oxbridge (maths graduates) and non-Oxbridge, specifically the Cambridge Maths Tripos, was that is was teaching the same content it had for the previous decades, whereas the maths courses at mine, and other 'Russell Group' universities had been dumbed down for the first couple of years. You could reach the same level as previous graduates by the final year, but you had to take a new additional course.