It was not just a few. When the vaccine came out you could see hundred of government officials claiming that the vaccine will stop Covid and that we will soon go back to normal life.
For example, here's Fauci on the 10.12.2020 claiming that herd immunity will be reached if 75 to 80% of the population would be vaccinated (at 0:45).
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Then, here's what we learn from Janine Small, president of international markets at Pfizer, on the 10.10.2022 at the European Parliament:
"Pfizer did not know whether its COVID-19 vaccine prevented transmission of the virus before it entered the market in December 2020."
Source:
Then, tell me how could Fauci make this statement about herd immunity in december 2020, since he had absolutely no data about transmission ?
So, TallGuyJohninBKK, if you want to call this "slightly overstating", it's up to you, but I don't share your point of view.