Because I have been extremely normal about this all day and have realized it's literally been more than twenty years and much of the fandom today does not in fact remember this, let me talk a bit more about the Mew trick and its history complete with ~archive links~!
Back in the late 90s and early 2000s, an endless array of playground rumours circulated about Pokémon Red, Blue and Yellow, in real life and on the internet. A few of them were real: going to the Safari Zone and then Surfing on the coast of Cinnabar really did cause Safari Zone Pokémon to appear, and if you talked to the old man who teaches you how to catch Pokémon and then Surfed on the coast of Cinnabar, you really would find a bizarre glitchy Pokémon called Missingno. that was a Bird/Normal-type, knew Water Gun twice and Sky Attack, and would give you over a hundred of the sixth item in your inventory.
The majority, though, were made up, just urban legends. Convoluted sequences of steps to obtain "Pokégods" or Mist Stones or get into Bill's secret garden. The most famous was the legend of Mew under the truck: if you went out of your way to skip the usual end of the S.S. Anne sequence by trading for a Pokémon with Cut or ensuring that you leave the ship by losing a battle after getting the HM instead of walking out, then you could supposedly Surf from the pier where the boat is to find a platform with a truck standing on it, and if you moved the truck with Strength, then Mew, the elusive Pokémon from the first movie, would be under it. The platform with the truck was a real easter egg - but it couldn't be moved with Strength, and there was no Mew.
By 2001, when I first found Mew's Hangout, the website that inspired me, it had a page about how to obtain Mew, which patiently explained that no, there's absolutely no way to obtain Mew outside of attending an official event or using a cheating device. All those legends you hear are nonsense. (The way to obtain Pikablu explained in the site's own cheats section, of course, was definitely real.) This had become the common wisdom by then: you can't get Mew, we know this by now, anyone saying otherwise is lying to you.
This is the context in which TheScythe posted a thread on GameFAQs on April 19th 2003 about how to catch Mew in Red, Blue and Yellow. He knew he wouldn't be believed, already on the defensive in the first post. A couple of people vaguely said they might try it for the heck of it, while others adamantly shut the idea down. It didn't get traction until another well-known user, Jolt135, posted confirming that it worked - and even then it was met with confusion and disbelief.
TheScythe hadn't discovered this trick himself; he explained he'd been seeing it on other websites, linking this page. That one is indeed another telling of the same trick, posted on March 9th 2003 - well postdating the site linked in my initial post here, where it was posted on May 31st 2002 (the submission time is not technically mentioned on that page, but it is on the actual list of tricks that was linked on Bulbapedia). It was rated 1.6 stars out of five. The trick had clearly been slowly making its way around the internet, but for nearly a full year at the very least it had failed to make any waves - because the Mew trick is just an incredibly fake-looking glitch.
When correctly performed, the only thing involved that even sounds at all glitchy is that there's a bit where your Start button doesn't work until you make a trainer walk up to you, which these early incarnations didn’t even mention, and then a bit where your menu pops open when you didn't press Start; both of those things sound more like the sort of random weirdness you'd today find described in a cursed game creepypasta than like glitching software. The Missingno. trick, the best-known glitch, involved garbled sprites and item duplication. This wasn't like that. Instead, it was just like the playground rumours: follow this arbitrary sequence of seemingly random steps, one that just so happens to probably be impossible to perform without restarting your save file, and then Mew will just appear. Nothing about this sounds like the game is being tricked into interpreting unrelated data in memory as a Pokémon encounter, even though that's what's actually happening.
Indeed, in TheScythe's thread Jolt135's first assumption was that no way was this a glitch, until another user, gamefreak19, tried it by battling another trainer instead of the Youngster and had a level 7 Machoke pop out instead.
Over the course of the next weeks, the Pokémon community studied what exactly was going on here under a microscope. This page was to my knowledge the first widely publicized partial explanation of what was going on under the hood and how the Pokémon species found was based on the Special stat of the last Pokémon fought. Even then, it had a lot of gaps (this predated the discovery that some of the "eight bytes on each side which all have the value 07" were actually the same enemy Pokémon's stat stages, and that by lowering its Attack using Growl, you can change what Mew's level will be, for instance).
But it was definitively a glitch, a pure accident of the game's programming. No employee at Game Freak or Nintendo could possibly have known this would happen and leaked it (despite the endearing way whoever posted the trick on JesseWorld used the username "Nintendo"). One of the millions of kids playing these games must have just happened to bump into it.
Probably lots of kids had accidentally triggered the Trainer-Fly glitch by just happening to press Start and fly away at the right moment before being challenged by a previously-offscreen trainer. Some of them were probably confused when they couldn't press Start or interact with characters anymore and restarted their game. Probably some kept playing and happened to get challenged by a trainer and then everything was fine after that. Probably some of them went back to the route they were on previously and had their Start menu pop up; some of them pressed B. And some of them would have gotten a strange encounter with a regular Pokémon and shrugged; perhaps some would have encountered a glitch Pokémon and been unsettled but ultimately also shrugged. You just had to have one kid out of those millions happen to get Mew, and then try to tell their friends about exactly what they did to get it. The exclamation mark while flying away from the Gambler probably registered as weird, provided a sensible starting point, let them replicate it...
And then they posted it online, and nobody believed them. Because why would they? It all just sounded exactly like all the other urban legends about how you get Mew. Nobody believed anonymous kids posting it on sites full of other made-up nonsense cheats; certainly not enough to painstakingly restart their save just to try it. The only people gullible enough to try it for a year would be other random kids - ones who were liable to mess up some of the convoluted, confusingly explained steps, who even if they did do it correctly and get their Mew would just become other anonymous kids posting it on other sites full of made-up nonsense.
I find it so tantalizing to think about this time: about the kids who had tried a Mew cheat that really worked long before anyone else but had no way of proving it and sounded exactly like all the others making up stories; about all the others (like me!) who saw their posts and just dismissed them out of hand because of course that's fake. We really do owe a lot to TheScythe for bringing it out of the darkness and into the light.