> You can absolutely analyze real world mass vaccination data in a scientific way. Just treat pre-vaccination and post-vaccination as two different patients.
Yes, you can do this. No, it's not the same.
With this kind of uncontrolled, longitudinal study, yes, some things are quasi-controlled: genetics, possibly lifestyle, other drugs, etc. Anything that plausibly doesn't change (much) in a single person from time A to time B.
Some things you can't control for, but still matter: changes in behavior due to the event itself. Placebo effect. Sample bias (e.g. the patients receiving treatment X were selected to receive treatment X because it was felt that they would benefit from it. This is subtle, but can really mislead. In the example of vaccination, imagine that the people most likely to vaccinate their young kids are also the ones most likely to keep their kids isolated at home, in a protective bubble...)
> So if your post-vaccination population has a rate of heart attacks 80% lower than the established norm, you know that is worth further investigation.
The key part of that is the last three words: worth further investigation. To get the final answer, you still need a controlled experiment.