Recent data ( https://www.frontiersin.org/journals/medicine/articles/10.33... ) suggests that any lives saved may only have been saved in the medium term;

"In the medium-term post-COVID period, the risk of death was reduced by 8% for those who had been vaccinated while in the long-term post-COVID period, the risk of death almost doubled for those who had been vaccinated. While in the medium term, there was a reduction in mortality risk for those who took two or three doses, in the long term the risk of death was greater for those who took one or two doses."