In the past "long term" side effects meant within weeks after the fact. This includes things like autoimmune disease. However, the statistical signal may only appear years later. The mRNA vaccines also don't "infect" the whole body, but only the injection site, and they are metabolized quickly. There really is no reason to assume any late effects say 5 years later.

If they were frequent late side effects, we would totally know by now.

By the way, they are a brilliantly less complex system, much easier to reason about, than traditional vaccines.