From Team Aqua/Magma on, all evil teams have been thorougly stupid, wanting to destroy the world for really contrived reasons. It mostly comes down to the hubris of their leader, who somehow tricks a bunch of grunts into following his plan. Said grunts follow the plan without questioning it, idiotically believing that "Team [name] will take over the world!" and that participation in the plan will somehow save them all from the destruction. They also dress in silly costumes, and serve more as comic relief than a legitimate threat.
And of course, the world as we know it is always at stake. This gets really boring after a while. It's always: "Save the world, 10-year-old, because nobody else bother to!". For being a threat to the world, surely the most powerful trainers around take the evil teams lightly. And true enough, the evil organization plotting to destroying/taking over the world is easily beaten by the 10-year-old. That gives me an impression that their plans weren't thought out at all, greatly diminishing the sensation of the world being threatened. In short, the evil teams from RSE on have been a bunch of weirdly-dressing doofuses who bite over way more than they could possibly hope to chew, with no consideration or no backup plan whatsoever.
Contrast Team Rocket, who, while having way smaller ambitions, also achieved the greatest success of all the evil teams.
Firstly, Team Rocket practically owned the region at the start of the game. Team Rocket members were bullying people and stealing Pokémon in every city in the region, save for Pallet Town and Pewter City (and maybe Fuchsia, I can't really remember). They weren't some fringe terrorist organization with greater ambitions than brains, they were the freakin' mafia. Their casino in Celadon raked in money for their organization, while also providing them with a hiding spot. Even though their lair on Cinnabar Island burned down, they succeeded in their plans there, creating Mewtwo (and subsequently losing control of it, but oh well, not really much they could do to stop that, and the player character had zero interference in that plan). The police were powerless to stop them, and Team Rocket managed to seize control of the headquarters of the greatest corporation in Kanto, stealing their technology.
And the grunts? Instead of brainwashed nobodies tricked into following a plan to the end of the world, they willingly joined for money and power. Team Rocket members, freely strolling the streets of Kanto in uniform, were openly bragging about their membership. Meanwhile, the name of the organization was only whispered by others, fearing the iron grip they held on Kanto. At the start of the game, Team Rocket might have been the single most powerful organization in Kanto, nobody opposing them but this one, idealist, talented trainer from Pallet Town. They were at the top of the game, having even infiltrated the Pokémon League (with Giovanni being an official Gym Leader), they were making butt-loads of money, and successfully stole all the Pokémon they wanted. And that was it. They didn't feel like needlessly destroying the world. Kanto was theirs already, they had got everything they wanted. They felt like a powerful enemy, because they were powerful. And instead of beating their leaders in silly outfits, you went from town to town in Kanto, beating back Team Rocket grunts, loosening their grasp on Kanto until you managed to convince their leader to give up and live a crime-free life. He then walks away, unscathed and unpunished. As a ten-year-old, you might convince him to disband the organization, but you have no way of giving him his deserved punishment, or making him speak for his crimes in court. Truly, Giovanni didn't fall from power, he simply stepped down and walked away a free man.
As for the legendary Pokémon, they stayed legendary. Zapdos, Moltres and Articuno were hidden in the most remote corners of the region, outside Team Rocket's sphere of influence, and outside their plans at all, for that matter. What could Team Rocket hope to achieve with them, anyway? They had the region at their fingertips already, no need to bother with a few ultra-rare birds hanging around in the middle of nowhere. Team Rocket didn't need any Legendaries to achieve their goal. Their goal was money and power, and they already had it all when you started your adventure.
Instead of simply observing an evil team trying to destroy the world (and being totally unable to influence it, apart from causing them to fail at the last minute), RBY had you face a powerful organization and actively bring them down from power. Team Rocket were way more cool than all the rest of the evil teams combined.