I am completely sympathetic to your motivations here. However, it seems like if you take that approach, where many qualifications are required before you are “eligible” to hold elected office, it’s much closer to an aristocracy than a republic or a democracy. Unelected elites who control the special university and those who set the eligibility rules in general hold and safeguard the real power, allowing trusted insiders to ascend to power and burying those with unorthodox ideas or worrying populist notions. It’s an anti-democratic idea.

It probably, on net, increases the power of elites and decreases the power of ordinary people, which increases the already extreme tilt in that direction. Like with a lot of things, it depends on the virtue of the elites in question whether this is a great idea or a catastrophic and oppressive one. As we can see, our current elites are so shameless they blatantly trade on privileged knowledge and can’t even manage to pass a budget for the country.