Moreover, if a pathogen is in general circulation, a person might easily be exposed to it many, many times in a few years, as you keep coming into contact with people who have it. I would be surprised if it caused the immune system to no longer respond.
More plausible would be that something about the vaccine itself might become less effective, since it is after all not an actual pathogen but rather a piece of one which we are attempting to "trick" the immune system into treating like a pathogen. I believe the term for this part of a vaccine is "adjuvant" (although IANAImmunologist). Most of the vaccinations appear to be mRNA based, if I am reading the appendix of the Lancet article correctly, so it is interesting to see evidence that this method remains "convincing" to the immune system (n=1).