So your model is just not historically correct in any point. Science didn't start with a few geniuses who then did experiments and then expanded to universities and so on. It's more that knowledge-generation (of different sorts) has been part of human cultures since the beginning, and what we think of as "science" is a specific way to look for knowledge about the natural world and it evolved alongside other aspects of culture over the course of many, many centuries. Education (like universities, which were one approach to doing education) was also something developing, but often did so with only a minor interest in what we would consider science today (basically astronomy and mathematics were the only "science" in the medieval curriculum; the rest were other things, and none were experimental). And universities were not major sites for scientific research until relatively recently (the mid-19th century).

OK, so how did you get to the larger scientific endeavors that we are familiar with today? The answer isn't academic science — it's industry. In the 19th century, there emerged industries that were based on scientific knowledge, and these ended up employing large teams of scientists, as well as creating partnerships with government and university institutions to create what we would think of as industrial research laboratories. The main place where this started was in Germany, where the chemical industry became the first "big" scientific industry in the world, and provided the template for many changes that would come later. This is also, non-coincidentally, where we start to see deep links between science, the state, and the military. The "German model" was immensely successful and was emulated elsewhere afterwards.

All of this happened alongside what we call the "professionalization" of science, when the idea that you could be a dedicated "scientist" came into being (the term was coined in the 1830s), and that you could go to school for this specific thing, specialize in something, and imagine having a reasonable slate of career prospects on the other end. This is very different than the situation in the late 18th century or earlier where to be a "natural philosopher" was still a very unusual situation and tended to be something that only high-class people would do (and you'd still have strange situations of "low-class" people being able to occasionally break into the field because of their apparent genius, like Michael Faraday, but they were the exceptions that proved the rule). Professionalization opened up the doors to many more people engaging in science, and the codification of many of the institutions of science we take for granted today.

Germany is an important site for this, and the origins of many of these changes. The Humboldtian Reforms in the 1810s, for example, created what we think of as the modern research university — universities, after this, began to transition from being institutions almost entirely dedicated to teaching, and into places where research was seen as the driving force (with teaching done to subsidize research through tuitions, and to provide the necessary "new blood" for the profession). The aggressive partnerships between industry, universities, and the state in Germany — again, pioneered by the chemical industry, but rapidly expanding to other fields of research — provided yet another template for how research could be yoked to the exponential engine of industrialization that had already transformed the world so much over the previous two centuries. The Prussian victory in the war of 1870 caught the attention of many outsiders, who noted that the industrial and technological aspects of it seemed to auger a very different world in the future, and led to reforms in France, England, and eventually the USA. Ultimately, a lot of what we see today in universities is a product of World War II and the Cold War, even more recent events, where the amount of government, military, and industrial support for scientific research increased by several orders of magnitude.

Anyway — this is just a sketch of the processes going on here, and one that is just limited to Western Europe and the United States. In other countries there are different stories to tell. Russia is a famous case of a state that attempted to harness these scientific forces differently, first under the Tsars (which appeared to chase the prestige aspects of science before thinking about the superstructure that produces them — Russia famously had an Academy of Sciences before it had a university!), and then under the Soviets (who essentially "socialized" science as a means of trying to accelerate and apply it to topics they were interested in, with results at times impressive and at times terrifying). China is its own different story (that I don't feel comfortable generalizing about!), and so on.