First of all, universities are not per se linked to the scientific method as we understand it nowadays. (Also, assuming that the scientific method is the only valid way to achieve knowledge completely ignores almost everything that is not a natural science. But that is an aside, because we're specifically talking about universities as an institution here.)
And I still argue that universities have not changed their fundamental nature, much as their exterior might have. Universities were founded as a universitas magistrorum et scholarium, a community of teachers and scholars (https://en.wikipedia.org/wiki/University), as they still are today. Already in medieval times, their two-fold purpose was to school the next generation of leaders and thinkers, and to provide a place for new knowledge to be discovered. That goal has not changed in any way.
Even when you go into further detail, things haven't changed as much as you make out. Funding continues to come from tuition fees, governments, and other big public organisations with a stake in the outcome of the universities' work. Their students continue to be mostly well-to-do young people aiming for a higher career; be that administrative, educational, theological, medical, legal, or academic. Their organisation in many places still revolves around the core concept of a group of professors, each in charge of their own research group (with attached students), and collectively in charge of the whole corporation. Other things too: the teaching forms of lectures and seminaries, the international community of university scholars, all that hasn't really changed in the last 1000 years.
Book suggestion: James Hannam, "God's Philosophers: How the Medieval World Laid the Foundations of Modern Science", is a good popular introduction into scholarship in the Middle Ages, and specifically talks about the rise of the university in this period. (https://www.amazon.com/Gods-Philosophers-Medieval-Foundation...)
And I still argue that universities have not changed their fundamental nature, much as their exterior might have. Universities were founded as a universitas magistrorum et scholarium, a community of teachers and scholars (https://en.wikipedia.org/wiki/University), as they still are today. Already in medieval times, their two-fold purpose was to school the next generation of leaders and thinkers, and to provide a place for new knowledge to be discovered. That goal has not changed in any way.
Even when you go into further detail, things haven't changed as much as you make out. Funding continues to come from tuition fees, governments, and other big public organisations with a stake in the outcome of the universities' work. Their students continue to be mostly well-to-do young people aiming for a higher career; be that administrative, educational, theological, medical, legal, or academic. Their organisation in many places still revolves around the core concept of a group of professors, each in charge of their own research group (with attached students), and collectively in charge of the whole corporation. Other things too: the teaching forms of lectures and seminaries, the international community of university scholars, all that hasn't really changed in the last 1000 years.
Book suggestion: James Hannam, "God's Philosophers: How the Medieval World Laid the Foundations of Modern Science", is a good popular introduction into scholarship in the Middle Ages, and specifically talks about the rise of the university in this period. (https://www.amazon.com/Gods-Philosophers-Medieval-Foundation...)