....we would need 80-85% of the population to have immunity. We could have (theoretically) reached that with vaccines.
Without them, that would have meant that we would have needed to have about 280 million people get the virus while we were shut down (of whom 2.8 million would have died, and maybe 20 million would have been hospitalized).
So, the answer to your question is no. The shut-down saved lives and had no effect on Delta.