What's confusing people now is the reduction in CDC recommendations about self-quarantine or isolation after a positive test or likely exposure. Used to be 10-14 days (can't remember when it went from 14 to 10). Now down to 5 days officially but even shorter for vaccinated folks in some jurisdictions or industries. That's based on evidence that someone who is vaccinated is no longer contagious after 2-3 days. By the time they test positive, probably already on the downside of the infection. Especially if they no longer have any symptoms.
The reason for pushing boosters is that someone who is boosted will have quicker response by protective antibodies as soon as the body detects SARS-CoV-2. Someone who is fully vaccinated but not boosted will have T-cells/B-cells that generate new relevant antibodies so they are very unlikely to get seriously ill but may have COVID-19 for a bit longer. Antibodies aren't designed to last a long time. When they aren't needed, no more are produced and the ones that are around naturally die off. All cells die and are replaced as needed. I learned that when researching bone density several years ago. Remember, I'm not a medical professional. Just a statistician with a background in clinical trials research who knows how to research a medical topic online from assorted sources.
