JX Valentine
Ever-Discordant
If I could jump in since I was the one who came up with the idea before then, minus Giovanni; it CAN be done if one had a good explanation for it. Like I made my Team Rocket good because the girl who wanted to reunite the team, did so in a certain executive's memory. However the only way that she could recreate her family and have them still named Team Rocket is if they changed their ways. Meaning this new formed group distanced themselves from their past and that's what she did.
While it's true that a good explanation would be needed on Team Rocket's end of things, one should keep in mind that people outside the organization aren't necessarily going to accept an organization that uses the same name as a crime syndicate just because its members insist they've changed their ways, especially at the beginning. It's going to take a lot of effort to get rid of that reputation, and those members risk having their past interfere with their present. (As in, let's say an entire town distrusts them because they're still considering themselves to be Team Rocket. That means for everything those Rocket members do, they'll be met with suspicion. Knowing the Pokémon universe and its fondness for vigilante justice, there might even be a few non-Rockets attempting to "stop" them from doing whatever it is the Rockets want to do -- including help people.)
So in the end, the problem is the exact one that wasn't entirely answered in the thread: the fact that Team Rocket has a terrible reputation. It would be far easier to create a fan-created organization that completely deviated from that image (as in, didn't wear Rocket uniforms, didn't call themselves Team Rocket, made no effort to look or act like they did as a crime syndicate), just because the public would bother them and get in the way of their efforts to do good in the world (or simply refuse to believe that they're good) otherwise.
This idea stemmed from when I was playing SoulSilver and Platinum and in SS, once I get to the mountain (names escapes me at the moment), a hiker bumps into me and apologizes then gives me the TM strength. I want to expand more on that in this story so I recreated that scene except it's with a Team Rocket grunt and a Team Galactic commander. The grunt is running to Chargestone Cave and then bumps into the girl, apologizes while she looks at him funny and then he gives her Strength. He explains what it does, actually demonstrates what it can do. However, I had it to where both didn't know how to use the HM given so our pink haired girl tries to take out her Teachy Tv etc. (oh and she fights her rival)
It sounds like it would make for an interesting one-shot, although there's questions here that I get the feeling would be answered by your general fic universe. Still, I have to ask: what exactly are the grunt and commander doing?
Actually, let me see if I've got this right. You're basing this off the event that happens in Johto (near Mt. Mortar), but you're having the grunt run to Chargestone Cave, which is in Unova. Is this actually taking place in Unova?
Then, there's the fact that you've mentioned Teams Rocket and Galactic. Are they working together here or no? Is this like that apprentice thing you were talking about months ago? (Also, yeah, why are they at Mt. Mortar/Chargestone Cave?)
Why does the grunt give the girl HM04? Just for looking at him funny? As an apology? Okay, sure, we can assume this is the good!Team Rocket that you're trying to build (although I'm not entirely sure when this is taking place in your timeline or if that's even true), but that actually introduces a couple more problems because if this is a standalone story, then you run into the issue that not everyone would know exactly how your universe works. (I have to admit, it's difficult keeping track of your stories for myself, just because there's a whole smattering of them and because the details of your timeline and its relationship to canon tend to change frequently.)
On the surface, it seems like it's a warm-and-fuzzy concept, but there's quite a few bits that don't entirely line up well. I'd say give it a bit more thought and try to answer all the "why" questions within your story. It would be interesting if you took the canon scene and tried to work things out so that the in-game hiker was a Team Rocket agent, but that would require a lot of thought, careful explanation, and closeness to canon.