To name a few:
In mainland China, most universties are public university. As a matter of fact, the introduction of private universities happened only a few years ago. So there's no need to add public or nation owned as part of the name.
Also when you name a university, usually you put its location first. e.g. 湖南大学，香港中文大学. This applies for foreign universities too. e.g. 加州州立大学 California State University, 亚利桑那州立大学 Arizona State University.
This is what it looks like now.
However, In the Republic of China, I mean ROC almost a century ago, not Taiwan, universities are divided into two groups: public and private. So, for easier separating them, you add 国立 at the beginning for public schools. They are funded by the ROC government and when ROC moved to Taiwan, the universities are moved too, with their names unchanged.
So what you are asking, is really, why people name the universities like this 100 years ago. I guess the reason, is because before that, the country was run by the Qing Dynasty and naturally the universities, if there were any, would be named like Imperial Technology University 帝国科技大学. 孙中山 wouldn't like this and want to emphasis that the universities are run by the republic government, not the empire. If you want your government sustainable, you start with education.
Fun fact, some universities in mainland China still has evidence of it's original 国立 name. For example