The CDC based that claim on incomplete and apparently misleading data
The Centers for Disease Control and Prevention is fighting Covid-19 without a full arsenal of data that some public-health experts said it would need to persuade more people to take steps to contain the pandemic.
www.wsj.com
"Inside the CDC, some officials disagree with the agency’s conclusion that vaccinated people who become infected may spread the virus as readily as the unvaccinated, and argue that more testing needs to be done, including tests that measure how infectious virus particles are, according to a person familiar with the matter
“There’s no one-to-one relationship between high viral load and infectivity, but we’re always making decisions based on imperfect data,” said Tom Frieden, who headed the CDC from 2009 to 2017."
Seriously, the CDC is a disaster and has been a complete mess since the pandemic started