I think you are mostly right about this except that the current crop of vaccines, especially as they wear off, don't offer a lot of protection from infection. They are what is called a leaky vaccine. They are really a prospective treatment to reduce symptoms.
I don't have it handy but another fairly large study (I think in Israel) indicated that prior infection and antibodies was 13x-20x more protective than the vaccines in preventing symptoms from infection. And I believe that study was from data pre and post Delta variant. It indicated that the antibodies from prior infection was as protective against the variants as the original infection. Makes me wonder if many of those infected were with the Delta variant which is why it was equally protective. Either way you look at it the data suggests that if you had a prior infection and still have antibodies you shouldn't need the vaccine. Even CDC appears to be headed in this direction which makes sense.