Perhaps believing that this was the case makes for a much more interesting world, and I'll just get disappointed when you prove this to be a silly myth/lie, or possibly even a PR stunt...
Proving a negative is notoriously difficult, especially for broadly-scoped claims. So you may not have to suffer disappointment per se.
That said, I am practically certain there has not been a Tamagotchi game where the game allowed only a single lifecycle of the Tamagotchi, and when (not if) the Tamagotchi died, the device would no longer function as a game.
At the time the game was released, I worked with a team that had a Tokyo-based partner, and I and all my teammates were given an original-release Tamagotchi as a goodwill gift by the partner. While I'm not sure where the game is now (probably lost in a box somewhere that was packed two or three moves ago and then forgotten on a shelf), my recollection is that I was able to restart the game after the current Tamagotchi life ended.
Again, this was an original v1 release of the Japanese market's game, which if the information you recall hearing had been correct, would be a single-play version of the game.
There were minor differences between the Japanese and US release, such as the displayed graphic on death (tombstone for the Japan market, angel wings for the US). Maybe the tombstone seems more "final" and thus led to the misconception that it was a one-time deal?
It really wouldn't make sense for them to sell a single-play device. The game itself was costly, approaching $20 (about $30 today), a single life could end in hours, and it was marketed for children. There weren't nearly enough children (nor are there now) with enough wealth that they, or even their parents, could afford to spend that much money for so little duration of enjoyment, to create a viable business model on a design like that.
Had there ever been a retail version of the game with such a user-unfriendly design, surely there would be some record of it available somewhere. And given the amount of actual documentation that exists today for the product, even the original versions, the lack of any mention of a "permanent death" release of the game is strongly suggestive that there never was one.
Again, lack of evidence is of course not proof of the negative. Proving this particular negative would require effort well beyond the scope of a reasonable Stack Exchange network question. But between my own recollection and the utter lack of any mention whatsoever of a release like that, I'm very comfortable claiming that no such release ever existed.
While I'm sure in your search of the Internet, you came across all these articles and more, I include links to the ones that I feel are either ones that would most obviously have mentioned such a release had one existed, or ones that are simply interesting for the historical perspective they offer, or especially those that enjoy both of those characteristics:
Hatchling Of Pet Lover Is the Rage Of Toylands (NY Times, September 1997)
Tamagotchi: Love It, Feed It, Mourn It (NY Times "On The Web", May 1997)
Tamagotchi (1996 Pet) (fan site wiki page for original release)
Tamagotchi Life Cycle (same fan site, life cycle overview)
Death (again, same fan site, page specifically about the death aspect of the game)
And just for completeness of course: https://en.wikipedia.org/wiki/Tamagotchi