A vaxxed vs unvaxxed study is:
Vaccinateded per CDC Schedule vs 100% UnVaccinated
That’s ridiculous. Though since your entire position is ridiculous, what’s one more thing? It’s a moving goalpost. The CDC schedule changes regularly. So if they did such a study ten years ago you’d say it was no longer valid, and if they do it now you’ll say it’s invalid as soon as the schedule changes.
But the whole premise is stupid, because it’s impossible that “vaccination” as a general concept could damage anyone. Any damage must be done by some individual vaccine, so it only makes sense to test that specific vaccine for whatever harm it is that it’s alleged to be doing.