The problem with that approach is that not everyone who gets a vaccine is fully protected. Some vaccines have an efficacy rate of 80% or even less.
What keeps these people from getting sick is “herd immunity” - the fact that the rest of the person’s social circle has been vaccinated means that the underprotected person will come in contact with relatively few sick people.
The more of the herd that is vaccinated, the lower the chances an underprotected person will get sick. So if you’re one of these underprotected people, you really want as many people as possible to acquire immunity.