It doesn't go against everyone having equal rights. It goes against everyone having equal power, which is not the same thing.

I'm also going to make a very bold and very unpopular claim that aristocracy is not an inherently bad thing. Every country already has an aristocracy of some sort, because aristocracy is defined as the group of people at the top of the social hierarchy. Even so-called communist countries have had aristocracies in all but name.

The only difference is that by acknowledging that you can't get rid of the existence of an aristocracy, you can begin to think about how one might control who is deserving of being in that class of people.

It is natural for intelligence to be somewhat tied to the education of one's parents. I don't see anything wrong with that. But at least with education, as long as people are given roughly equal educational opportunities, there will be chances for social mobility, and much more so than today. If you take a look at China's imperial examination system, as flawed as it was (largely based on the arbitrary memorisation of Confucian classics and essay-writing), it still provided unprecedented social mobility for the time, where any literate peasant could obtain a well-paid job in the imperial bureaucracy and prestige for their family. Yes, already-educated people had an advantage but that is not necessarily a strictly bad thing, as unfair as it seems from first glance.

Let me give a scenario to think about (this is not a proposal but just some brain food): What would happen if we administered a university entrance exam to all seekers of legislative office and gave the positions to the top 100 highest scorers? Obviously the average rich person would have an advantage over the average poor person, because they have better educations, but at the same time, poor people would have a much better shot of actually getting the office than they would under a purely democratic electoral system, and we have the important benefit that whoever does get the job is far more likely to possess basic thinking skills.

Again, not a real proposal, just something to think about. The system described above would definitely suck in reality if implemented as written, and it doesn't stop smart but malicious people from obtaining power.