What's untrue exactly? Here's a review of more than one study that found reduced infection and transmission potential. The results aren't as clear cut as those for preventing severe infection because it's much more difficult to study transmission (ethically and logistically) in a robust method.

Also, when I said it would be really weird for a vaccine to reduce disease severity and not transmission potential, I mean that there's no known mechanism by which that can happen. A vaccine can't reduce only the virus particles that stay in the body and skip the ones destined to be expelled. The immune system doesn't work that way.