That is my understanding as well, but it is entirely irrelevant to the point being made. If someone on the anti-vax margins get the vaccine and proceeds to get sick and test positive, the are more likely to say "the vaccine doesn't work" compared with "I would have gotten more sick if I wasn't vaccinated". You also wont here all the stories of people who got the vaccine and proceeded to not get sick; so there is a sampling bias to hearing about cases where the vaccine doesn't work as well.

The concern here is about the longterm PR of vaccinations, not the technical benefits of the vaccine.