I don't think an appeal to authority is gonna work in this thread.
While Plato is something of an authority (and thus the best target for the joke), he does show his work. HN is pretty good about avoiding appeal to authority.
But it is a frightfully common logical fallacy, and the perhaps the only one we spend so much time educating our children to use. Largely by example.
All logical fallacies are actually valid ways of thinking about things.
It's good to avoid them (or use them) but only in moderation.
The idea that avoiding logical fallacies all the time is OK, is actually illogical.
Sure, something might not be true just because an authority on the matter said it, but it's more likely to be true than what some random person said about the subject. E.g. one would rather trust Einstein than the guy who says they invented perpetual motion. The idea of the burden-of-proof (which is totally valid) is against total adhesion to "appeal to authority".
And given finite lifetimes, and other interests, we're not gonna all learn medicine and evaluate any claim on a medical issue we have from our street cleaner and our doctor alike. We'll trust the doctor, appeal to authority be damned -- and it's the smart thing to do, even if it's not 100% accurate (the doctor might be wrong).
In fact conversation would be 100% impossible without "appeal to authority" of some sort, except if you expect anyone to prove everything from first principles.
The quote is a quote from a well regarded (for millennia) piece of thinking on pedagogy. You're not asked to take it as truth because Plato said it, but it's worth to ponder what he had to say, especially since it's a foundational part of western civilization.
You can read it as "hey, millennia of people though this has a good point -- maybe there's something into it?". Or at least at "hmm, here's what many people used to think is a valid point about pedagogy".
But it you're turning off your logic in casual conversation, expect to get burned. Detecting fallacies should be an always-on feature, like HTTPS. Most of the time you're not going to have a MITM attack... until you do.
I think you've also missed the point of TFA, which says that anti-authoritarians can accept authority, if it proves to be a legitimate authority. And it still requires you to use logic, because no authority is perfect. I may be a Lutheran, but that in no way makes me a creationist like Martin Luther.
I don't suggest turning off logic in casual conversation. Rather I suggest that one doesn't put it in autopilot -- as if fallacies (or their lack thereof) are the ultimate judgement of an argument.
I too read for the freedom and power it brought me. So many of us, I imagine...