Actually there already have been previously undocumented side effects with the new vaccines, heart inflammation with the mRNA vaccines and blood clotting with a couple of the others. These were the short term side effects that weren't caught during the safety trials. We have zero data about long term side effects.
While I can't give you an "actual pathway to a long-term problem" by which I assume you mean a mechanism of action I will note science is hard, scientists can't always predict what will happen, and serendipitous discoveries happen daily. The long term risk is unknown. I think if you're at risk for serious infection you should get vaccinated, if you're not really at risk it's more difficult to weigh the relative risks.
While I can't give you an "actual pathway to a long-term problem" by which I assume you mean a mechanism of action I will note science is hard, scientists can't always predict what will happen, and serendipitous discoveries happen daily. The long term risk is unknown. I think if you're at risk for serious infection you should get vaccinated, if you're not really at risk it's more difficult to weigh the relative risks.