Here's the rub: is previous infection as good as Moderna? Probably not, especially given the new variants. (Though last year studies were showing infection providing 90%+ immunity.) But nothing is as good as Moderna. (Except maybe Novavax?) Should we require J&J vaccine recipients to be re-vaccinated with Moderna? Nobody is demanding that, though very roughly speaking J&J-level immunity may be about where things stand now for the previously infected.
This seems to be another unforced err by the CDC. Not as bad as some others, but unforced and very politically costly. It also complicates things going forward because it seems we're quickly approaching the point where we'll need to begin requiring a 2nd round of vaccinations. The CDC will need every last shred of credibility and good will left for that initiative. It could have been in a position where it could say, "hey, we tolerated the previously infected skipping vaccination before, but now the situation has changed." Instead, they'll have much less capital to work with.
I was trying to be conservative. I don't have any hard opinion on this one way or another. It seems like a degree of precision we'd be lucky to achieve in the best of times--i.e. when the virus and demographics aren't constantly evolving. And it's a degree of precision that is unnecessary to establish in this particular debate, even if we could. It feels like bike shedding.
The question isn't which is better--infection or vaccination. The question is, do we demand that the infected be vaccinated even though the substantial and consistent weight of scientific evidence gives us little reason to believe the previously infected put the population at more risk than the vaccinated, especially relative to vaccines that we affirmatively accept as sufficiently efficacious? The only scientifically justified answer is an unqualified, "No".
There are innumerable other questions people can debate, but from a pandemic policy perspective there's no need to venture there. And, frankly, it seems imprudent considering the level of uncertainty surrounding everything--not just the quality and relevance of data, but even if you're technically correct one moment, things can go sideways the next moment, necessitating a change in policy.
The CDC study you're pointing to says that among people who have been previously infected, those who have been given vaccines are less likely to be reinfected than those who have not.
There is no error from the CDC here, nor any lost credibility.
With the absence of evidence in either direction, it is unwise to draw either conclusion.
An alternate conclusion to your own is that people who get the vaccine did so in order to take advantage of the reduced restrictions on vaccinated people at the time.