I believe the news report you cited is not accurate in its reporting. Of course, it's not actual science, and we will wait for the release of the actual science, but here's my understanding (which may end up being wrong):
if vaccinated people get infected anyway, they have as much virus in
their bodies as unvaccinated people.
This seems to be incorrect. My understanding is that the CDC info looks at vaccinated people who were infected AND developed COVID symptoms, not all vaccinated and infected people. Also, the tests did not look at the virus levels in the whole body, just in the nose and throat. Obviously, the nose and throat release virus into breath which is exhaled, so that directly impacts how contagious a person might be. However, what should be important for the person's own level of illness is the amount of virus in the lungs and other internal systems, not the nose and throat.
This mis-reporting would seem to account for the apparent discrepancy you noted.