I don't really buy this line of reasoning.
Educated countries make baffling, crazy choices at the ballot box all the time. Plenty enthusiastically elect "one person, one vote, one time" types of parties, or something close to it. For example, Russia is fairly educated if one believes university graduation numbers. Poland and Hungary are more "educated" than Austria or Czechia. And all of them are so much more educated at every level than Portugal, which has had a string of competent and non-authoritarian governments. Handy coloured maps[0][1] offer little in terms of "ah, I see how the ones who send their young'uns to uni and spend the most on education are the most competently governed".
There's probably some correlation between education quality/amount in very specific areas, like civics and basic economics, and "quality" of elected governments. But I intuitively believe (admittedly without searching for evidence) that these are easily swamped by more important factors, like economic/political crises du jour, history, culture, neighbourhood, etc.
(regarding neighbourhood, I think this is often underappreciated. E.g. maybe if Portugal was surrounded by third-world tinpot dictatorships, it would also join that club, but it's influenced far more by countries like Spain and France; and it's very difficult for an almost-out-of-the-abyss country like Ghana to break out and join the club of reasonably-governed middle-income countries given its neighbourhood).
[0] https://ourworldindata.org/grapher/government-expenditure-ed...
[1] https://ourworldindata.org/grapher/share-of-the-population-w...