I mean, technically you’re correct that the proof was not valid and the counter-example just makes that more obvious but also mathematics is built on axioms and proof chains that we’re pretty confident are all valid but all are up for being taken down. That’s the thing, even with something like pure mathematics then it’s far from easy to see a tricky flaw, especially if the proof isn’t valid but the idea is in fact a theorem (ie the thing you’re trying to prove is true, but you just haven’t actually proved it - there cannot be a counterexample but also someone can show your proof is incorrect).
I should probably have been clearer that ease of disproval is relative and non-uniform. Some things are incredibly easy to disprove via a trivial example and so there are lots and lots of things we know are not true. But as we slowly build these towers of ideas then counterexamples become rarer and even being certain about a flaw in a proof becomes the effort of years of work a lot of the time. Sometimes counterexamples are wrong, but it’s somewhat rarer because they usually make it easy to understand how the idea cannot be true (but to do so you probably want a minimal counterexample and then we’re into a whole other area of hard work).
I think mathematics is a great example when discussing knowledge/the scientific method/understanding (partially because 15 years ago I was doing it at university so still haven’t quite forgotten everything I knew back then) because it is held up as something so pure and always building. It’s only when we move to things like climate change that bad actors attempt to “teach the controversy” in a way that goes against the scientific method. But we don’t need to start there as mathematics has plenty of examples of how it’s not a perfect straight path and sometimes things need to be re-evaluated even without competing ideas.
The greater point is that a lot of ideas are easy to know are bad and that doesn’t take away from how grindingly slow progress is and how everything we do know is not perfect so there are areas of valid debate. But those areas do not include things which can easily be shown to be false - areas where someone without domain expertise may not know enough about the terrain to even know the difference between something obviously false and an actual point of academic dispute. There is a huge issue with people who do not know enough to know what they do not know being considered important voices needed for debate (simply because conflicts make for good theatre) - this runs counter to everything we have achieved in building these towers of knowledge. The understanding of the variable difficulty of proof, disproof, etc helps to prepare us to spot when we are sold controversy (and a claim of potentially equal validity) rather than good ideas.