Your arbitrary criteria

I am contesting your criteria. Your criteria are not relevant to my original claim.

1. Your requirement for Dutch originating sources: Belgium is trilingual and the language regions may predominantly be separate, but the local culture is shared and/or overlaps. Just because the Flemish speak Dutch doesn't mean that their entire culture is rooted only in Dutch-speaking traditions.

   Historically, the region known as Belgium today has been part of both the French and Dutch empires (and, by extension of the Dutch, even the Spanish empire), which means that part of our cultural heritage is French and Dutch. French and Dutch cultural history from before or during the time Belgium was part of that empire are fair game when justifying or referring to Belgian culture.

2. Your requirement for printed fables which use the explicit phrase. I didn't not state that Flemish fables used it, nor that it was used in printed fables (it is the case that it originated from a fable, as proven below, but I did not claim it to be the case).

   I stated that the expression is used in Flemish, meaning "a long time ago". That may include fables but is not strictly limited to them.
   Regardless, I added ample evidence of its usage, in several cases with an explicitly mentioned context of fairy tales, alongside the alleged origin of the quote from a French fable by Jean de la Fontaine.

   Even by the Korean context that sparked this challenge, "folk tale" is not the same as "printed fable". And again, I didn't even claim that it's used in a specific folk tale.

   Also keep in mind that fables in print are easier to historically track, but it's not the only way in which they are passed down. From personal experience and that of three friends I talked to today; grandparents, great grandparents and kindergarten/primary school teachers used the expression when telling stories.

3. I have no idea why children should be the target demographic according to your criteria. Historically, fables have not been solely targeted at children, so the requirement is nonsensical.

4. In the above examples, the phrase "when the animals could still speak" refers to a time period that is at most a few decades ago and introduces customs and circumstances that may seem quaint to young readers today.

   Time is relative. In the examples you and I both refer to, it's used tongue-in-cheek to pretend like the period in focus was a very long time ago.

   And that's the point: the expression explicitly means "a long time ago", but is comically used in cases where the time period in focus is relatively recent. Overstating something is a very common and casual form of comedy.

5. User @Tonny, who lives on the Belgium-Netherlands border and who has many relatives in Flanders, says that they have never heard this fairy tale opening; the standard opening is "Er was eens ..."

   I never claimed "toen de dieren nog spraken" was the only or predominant introduction to fairy tales. I agree that "er was eens" is more prevalent, partly because the Netherlands Dutch region exclusively uses it (as far as I'm aware).

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Usage in Dutch/Flemish

1. Flemish dictionary

   vroeger toen de dieren nog spraken (uitdr.)
   heel, heel lang geleden

   My translation:

   In the past, when the animals still spoke (expression)
   a long long time ago

2. It is also the name of an event centered around storytelling.

3. This article uses the phrase specifically to mean "a long time ago". It's behind a paywall but if you disable scripts you can read the relevant part:

   Het was nochtans geleden van toen de dieren nog spraken dat we hier nog eens een echte wolf mochten verwelkomen. In een paar jaar tijd werden het er drie. En dit voorjaar bleek wolvin Naya zelfs zwanger: het zou de wolvenpopulatie in ons land verdubbelen.

   Google translated with a minor improvement from me:

   It was back when the animals spoke when we would welcome a real wolf here. In a few years it became three. And this spring, she-wolf Naya even turned out to be pregnant: it would double the wolf population in our country.

4. A 1947 collection of fables uses it as a title. The foreword focuses on fables and specifically points out that animals/inanimate objects are the actors in the stories.

   [..] het vertellen van een verhaal, ontleend aan dieren- doch ook wel aan planten-leven, welk leven door den verhalenden dichter min of meer wordt ver-menschelijkt.

   I didn't translate this as it's an older variant of Dutch and quite poetically phrased, Google struggles. My above summary is an accurate translation of the relevant part.

5. This site uses the expression to herald back to the "ancient time" of when their company was brand new:

   Ik neem jullie even mee naar de beginjaren van mijn bedrijf. Toen de dieren nog spraken en ik nog geloofde in ondernemers-sprookjes.

   My translation:

   I'll take you back to the first years of my company. Back when animals still spoke and I still believed in entrepreneurial fables.

6. This site uses the expression and specifically points out that the expression is used in the context of a fairy tale:

   En het busje kwam. Een echt prachtexemplaar. Maar wel een uit de jaren stillekens. Om in de sprookjessfeer te blijven : uit de tijd toen de dieren nog spraken. Niks geen vering. Er was een stuur en er was ook nog een gaspedaal en daar moesten we al dik tevreden mee zijn!

   My translation:

   And the minivan came. A beautiful minivan. But from the old days. To stay in the context of fairy tales, back when the animals still spoke. No suspension. There was a steering wheel and an accelerator and that's what we had to settle for.

   At this point I'm just quoting Google so let's leave it here. Google "toen de dieren nog spraken" and skip the first page as it mainly refers to the event (my second usage case), not the phrase itself. Starting from page 2, most search results are original usages of the expression.

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Origins

The quote seems to originate from a fable by Jean de La Fontaine, who is famous for writing animal-centric fables reminiscent of classic Greek/Roman natural analogies (Aesop among others).

The French translation is "le temps où les bêtes parlaient". According to this site, it's pulled straight from one of de La Fontaine's fables:

Du temps que les bêtes parlaient,
Les lions entre autres voulaient
être admis dans notre alliance 

Reference: Jean de La Fontaine, Fables, iv, 1 (1668), éd. Marc Fumaroli, Paris, La Pochothèque, 1985, p. 197.

I cannot personally confirm if this quotes the fables, but multiple sources claim it is a direct quote when you Google those three lines.

The site I also delves into how many cultures have fables revolving around anthropomorphised animals. I Google translated part of the linked page:

Those who lived their childhood in languages ​​other than ours were able to learn to name this time, pretty much whatever that language was. While this universality of the animal apologue has been noticed at least since the eighteenth century, it is strange that it still has not, to my knowledge, been the subject of a systematic study of this universality, which means that animals speak not only in the languages ​​of Europe, but also in those of the Chinese, Berbers or Samos, [..]

Also note the definition of an apologue:

An apologue or apologis a brief fable or allegorical story with pointed or exaggerated details, meant to serve as a pleasant vehicle for a moral doctrine or to convey a useful lesson without stating it explicitly. [..] The term is applied more particularly to a story in which the actors or speakers are either various kinds of animals or are inanimate objects.

This website, which coincidentally also mentions the Korean tiger-smoking phrase that sparked this discussion, attributed the speaking-animals phrase to Catalan culture, but since I can also find references to Dutch and French usage, it seems to be much broader than only the Catalan region.

Note that all instances I've found so far are relatively close to France, which makes sense if the quote originated there. The regions spanning from Spain to the Netherlands have also historically been part of the same empire (Belgium-France, Belgium-Netherlands, Netherlands-Spain), so the usage may possibly have transferred from being the same empire (temporarily).