I think that whatever we do should have a legitimate face to avoid close scrutiny from Giygas-controlled authorities and earn cash to fund our secret fight against evil aliens.
Eeeeh...
Legitimate face shouldn't be for
Giygas. If Recerd is any indication, we're probably already a reasonably known quantity. Hell, we were one of the very first people attacked by a Possessed Object and we saw Giygas in our dreams at the start.
Even if they don't know... we should probably assume that Giygas will EVENTUALLY learn about us and move against us.
However, remember -
Giygas' plan means he has to act in secret. So if they move against us, they must also move in secret.
Having a front - and I'd argue it doesn't actually need to be
reputable, it just needs to not be "HEY WE KILL ALIENS WHO WANT TO GENOCIDE EVERYONE!!!!" - will help to maintain this as a Shadow War instead of encouraging Giygas to start to operate openly.
This being a Shadow War gives us a significant advantage in terms of what Giygas can't do - which given the thing has armies of Starmen and advanced ships, is a LOT. Outright war would be more risky for them, given what would happen if the City did unite in the face of such a threat, but given the sheer power Giygas posesses... I'm not sure it would be a clear cut or easy win.
It also means that we can more easily get our actual goals done under the noses of people who are working for people who work for Giygas, but who don't know about the Shadow War, and thus won't know what we're actually trying to do or how we do some of our things.
Edit: Again, I don't think it needs to inherently be a 'neat' public face. If we're a criminal group, but our goal is seemingly more reasonable and mundane in comparison, then people not already in the know aren't going to go looking
I wanted to do KND stuff in order to capture the energy of the kids in Onett against the Fresh Breeze movement, and given that this kind of war is going to hit kids who can't defend themselves especially hard (especially other kids like CD with family issues), then a Youth Protection League would feel like it could expand. A criminal Org, since it's inherently anti-authority, but one that manages to feel rebellious without also being too open and keeping some of that KND vibe I'm going for.
...Actually, you know what? I think you could combine the Aesthetics by combining the dual meaning of the word 'League'. Like, I'd still make it criminal (I.E. Against the Status Quo) on principle, but also emphasising that it's for providing a community and sense of purpose and using sports to help that might work.