So people lose trust because of spin and "do their own research" and it's vaccine positive messaging (about the vaccines that are safe and work well) that is ultimately the problem?
There is no one problem (nice strawman), but when the CDC makes bold claims in a chaotic, dynamic environment that are unjustified based on the science, which they have done endlessly, it erodes trust to the point where people presume they are bad faith, have conflicts of interest, or are simply incompetent, so subsequent messaging has less effect.