The latter.
The problem is less about whether the world supports the ability of individual members to exist ( as you point out, it does ). Its that the story of that world doesn't have room for their existence to have meaning. Core to the premise of the Teen Titans is that they are young, new heroes, a new generation following the old. Even if you go with a roster that's not literal sidekicks, this just doesn't work when they are outright contemporaries of the Justice League. As things stand, the Teen Titans would have almost the same level of experience at being a hero team as the Justice League, and that just doesn't work.
Combine that with the Cyborg problem ( one of their core members having been preemptively yanked ), and the oddity of Dick Grayson recruiting them all when Batman just recruited the JLA himself, and you have a frankenstein's monster composed of Teen Titan parts somewhat shambling along.
Basically? For Teen Titans to work, it really needs to happen a goodly number of years after the founding of the Justice League.
Maybe my perspective is skewed from being both more of a fan of G4 characters like the Young Justice roster and having been exposed mostly to the cartoon version of the New Teen Titans roster, but I don't value the experience gap between these characters and the Justice League as much as you do. Plus, I feel like the Titans are the quickly accepted young heroes; generational clashes are always more of a Young Justice thing. And o kind of feel the public's awareness of the property justifies their existance as much as anything else; Cyborg is still regarded largely as a Titans character even among mainstream folks, so go ahead and give the people what they want.
The NTT kids existed pretty well in a seemingly League-less world in their cartoon, easily the most mainstream variation of the team. Their youth was still a massive factor in their portrayal, don't get me wrong, but they didn't need any senior team to exist above them for the sake of the show. Plus, Batman, Wonder Woman, and possibly Aquaman all seem to be veteran heroes to whom the contrast you desire would be evident. I do kind of agree with the idea that having Cyborg and Nightwing likely being "upgraded" may impact the team's chemistry with the overall world, but not to a debilitating degree.
Plus, I've never been a major fan of the idea that a comic character needs a purpose; it certainly helps but sometimes I want to read a Tim Drake story because it's Tim Drake, even if he 's no longer Robin, can't be Nightwing or Red Hood. And I kind of think the team could serve a similar purpose to Nightwing's; to show how the next generation of heroes, even if they're only a bit younger than some Leaguers, have a better relationship and foundation to stand on a s the next defenders for he Earth. Snyder's Justice Legaue seems to have formal and professional feel to it; have you ever heard of a Titans team where the author didn't even try to make them the club of friends?
To me, we resolve your issue by emphasizing their comparatively young ages in comparison to the League, make their team chemistry radically different and warmer, and go ahead and make them the teens to the Justice League's adults. And then you keep expanding they number of heroes; I want to at least see Wally on the big screen before I die, and I'd love to see Tim, Jason, And Damian.
