I don't know about the rest of your methodology and I'm suspicious of the concept of "Oh they forgot to multiply by 13%" cause you just don't multiply percentag

I don't know about the rest of your methodology and I'm suspicious of the concept of "Oh they forgot to multiply by 13%" cause you just don't multiply percentages (with very few exceptions). But more to the point, the group "Covid Positive + No Vaccine" has a high sampling bias built into it. Because you're sampling "People who lived through covid without the vaccine".