So if you have typically 99.97% recovery rate, and the "vaccine" has a 95% (in theory, in practice ~60%, check Israel) efficacy, then after needle you would have a 99.9985% ((1-.9997)*(1-0.95)=0.000015, 1-0.000015=0.999985) recovery rate. So we're talking about a <0.03% difference. Why would anybody do anything remotely controversial for a 0.03% benefit? Why would anybody expect to see differences in the population from a 0.03% benefit? And that is just strictly talking about the benefits in regard to COVID without regard to any negative consequences.
Math is fun!
This looks right, probability of non-recovering naturaly, times probability of the vaccine failing, yields new probability of non-recovering.