It's just that the nuance is often ignored in the initial saying. Why do people try to debunk something out of a game of telephone instead on restoring the original information.
Point has always been that people who are less ill, less severely ill, for a shorter duration, with fewer or no symptoms, have lower chances of infecting others.
The urge to simplify is great but it causes these weird discussions like "does the vaccine prevent (all) infection" or like what we have here.