Since this question arises, and specifically because the answer is complex, I thought it might be worthwhile to look at why immunization is so important.
Short answer: YES!
I remember a conversation I had, years ago with a successful businessman who had grown his business after having dropped out of college in his first year. He felt that since he had read the books, his education was as good as anyone else’s. This statement made me examine the value of an education, particularly one in the “hard” sciences. Was my education simply a matter of reading the books, memorizing the vocabulary and learning the chemical reactions? Is education a matter of acquiring a set of facts and being able to recall them when needed?
I came to the conclusion that education is far more than that. It is the teaching, mentorship and coaching on HOW to read the books, HOW to apply the facts and HOW to use those facts to better understand what is and what will be happening.
This is true for whatever you do. The simple acquisition of a response to a stimulus is insufficient to be prepared to future challenges.
The same is true for your immune system. When you become infected with a virus, you mount a shotgun response against whatever your system can see. Take for example COVID. When you become infected, you mount an immune response to all of the proteins on the virus surface, and all areas of the Spike protein. Empirically, if you have recovered from an infection, the net cumulative effect of your response has been sufficient to protect your vital organs from damage that might have proved fatal.
There is a very important difference though between a clinical immune response to a viral infection and a stimulated immune response to a vaccine, and that difference is why it is critical to add a vaccination to your immune arsenal.
Since your immune response is not “learned” but generated from a pre-existing random set of antibody specificities, the response to an infection is also composed of a subset of your randomly generated antibodies. Recovery from the infection may have been due to an immune response to many different things, but not necessarily to the binding region on the tip of the Spike protein, and it is that subset of antibody specificities that will prevent you from becoming infected again, and for which variant mutations may escape.
Vaccinations are specifically tailored to “find” the subset of your own personal immune response repertoire which will allow you to be protected from subsequent infection and make sure that it gets stimulated and prepared to fight off those infections.
That is why it is so important for EVERYONE to become vaccinated whether you have seen the virus before or not (and whether you actually know whether you did or not).