When people reference the CDC not being able to study gun violence, they're referring to the Dickey Amendment of 1996. With the ambiguity in interpretation of "none of the funds made available for injury prevention and control at the Centers for Disease Control and Prevention (CDC) may be used to advocate or promote gun control" coupled with an earmark that shifted the exact amount the CDC had allocated to gun violence research to something unrelated, it ended up being, at minimum, a de facto ban on gun violence research (with a strongly implied threat of funding cuts for breaking the ban). There were exceptions to the de facto ban when they had marching orders from elsewhere, e.g. Obama's directive after Parkland.
That said, 2018 finally saw an interpretation of the Dickey Amendment codified that allowed, unambiguously, research into gun violence. And 2020 finally saw funding allocated for gun violence research. So, after over two decades, the CDC finally regained both the go ahead and the funding.
Given the above, I think it's still reasonable for people to use the shorthand of saying the CDC had been banned from doing gun violence research. Softening the language to be slightly less absolute would help, but I'd consider people to be in the ballpark of correct when they call it a ban on research. And it's also correct that that's no longer the case, thankfully.