Take a look at the vaccine studies - only J&J actually tested the effect on transmission, and J&J was hardly used.
The Pfizer and Moderna vaccines never tested transmission. So why would Fauci make such a claim?
Pfizer trial protocols: https://www.nejm.org/doi/suppl/10.1056/NEJMoa2034577/suppl_f...
Moderna trial protocols: https://www.nejm.org/doi/suppl/10.1056/NEJMoa2035389/suppl_f...
But what did Fauci say on national news?
“So even though there are breakthrough infections with vaccinated people, almost always the people are asymptomatic and the level of virus is so low it makes it extremely unlikely — not impossible but very, very low likelihood — that they’re going to transmit it,” Fauci said.
Appearing on CBS’s “Face the Nation,” - June 16, 2021