The mRNA vaccines are biochemically quite simple.
mRNA vaccines have been studied in people for ~10 years.
The whole point of mRNA vaccines is that they are easy to design and deploy
We generally have a good understanding of how RNA behaves in the cell. A KEY feature is it's short half life! The mRNA from your vaccine will be gone in a week or two. The expressed proteins in a similar time frame.
There are plenty of things we adopt enthusiastically without decade long large cohort studies. To require those for every human-interacting piece of tech would mean we make no progress.
We can use well-informed priors backed by decades of research to make good predictions about long term effects.
In this case we understand mRNA tech quite well! We understand the immune response to coronaviruses pretty well too!
Treating every new tech as a black box that must be full studied for a century for deeming it 'safe' is an overzealous application of the precautionary principle.
Doing something like cutting your driving by 10% would have a much larger effect on lifespan, for example.