I mean you can literally witness it with your own eyes right now when you look at the US on the world stage. Compare the quality of governance and selection of candidates to a country like Singapore, or hell even China whose ruthless political apparatus weeds out people who don't perform quite effectively.
The US's strategy of funnelling the power through visible, legal channels with some levels of consistency is by far the best. If they want their kids in prestigious universities then that is what will happen. There isn't an alternative where the powerful shrug sheepishly and let their position erode.
Let's tax the rich more and give everyone access to higher education, even the playing field out. Yes you can redefine what rich means, and what poor means, over time through policies.
That last part sounds like almost a threat regarding what rich people might do if you block their corrupt behavior. That's already a sign there's a problem with the power dynamic here.
Someone getting in by buying the institution a new building isn't denying anyone access to higher education. In fact, since the institution has a new building it is likely it is increasing access to either education or research opportunities.
> Let's tax the rich more and give everyone access to higher education...
30% of the US does get a degree, so higher education isn't exactly an elite marker.
[2-4 years of time] X [70% of the US that doesn't have a degree] is an enormous number of hours. If someone is not going to use it then they have better things to do than academics. The risk of most of that 70% just looking back at a couple of wasted years is pretty high.
I suppose in fairness it might work out, but even now there are a lot of people getting degrees who havn't managed to parley it up into better outcomes. But it is likely to be a terrible plan, expensive and with marginal positive outcomes.