This claim is frequently made by people trying to compare dissimilar cities, states, countries, regions, etc. without any attempt to adjust for those differences. More importantly, they are making an assumption that the curve wasn’t flattened, thus ignoring the reality of the counterfactual. This is a common mistake that is made when looking at prevention activities. People complain about the interventions whether from a cost or inconvenience standpoint. In my field, we are used to that, and are often looked at as a cost center instead of as a cost savings center, which can be empirically shown. We end up getting viewed as “why are we funding this department” until things go south without our help, and then they wonder why they didn’t have more resources dedicated to our efforts. It’s a constantly swinging pendulum.
Some obvious evidence for how they helped is obvious when looking at US cases compared to stringency, which is a measure of those measures.

Imagine if wave 3 had been where the first wave was. It would have been devastating to healthcare given the lack of PPE and beds. Think of the experience of NYC across the entire nation…AT ONCE. In fact, it would have been even considerably worse than what wave 3 is here. It’s possible to use some calculus to predict the curve based on a number of different variables and would likely have gone completely off the chart space above. Given that the entire population was immune naive, unmitigated spread would have devastated lives and the economy.
“By comparing the deaths predicted under the model with no interventions to the deaths predicted in our intervention model, we calculated the total deaths averted in our study period. We find that across 11 countries 3.1 (2.8–3.5) million deaths have been averted owing to interventions since the beginning of the epidemic.”
“The modern understanding of infectious disease, combined with a global publicized response, has meant that nationwide interventions could be implemented with widespread adherence and support. Given the observed infection fatality ratios and the epidemiology of COVID-19, major non-pharmaceutical interventions have had an effect in reducing transmission in all of the countries we have considered. In all countries in this study, we find that these interventions have reduced Rt below 1, and have contained their epidemics at the current time. When looking at simplistic counterfactual models over the whole epidemic, the number of potential deaths averted is substantial.”
“Our results show a strong and statistically significant correlation between New Death Rate, New Case Rate, New Mild Case Rate, and the treatments. We portrayed the variation and captured the “status shift point” of PT [Pharmaceutical Treatments] at various levels of NPT [Non-Pharmaceutical Treatments (also known as NPIs]. Crucially, the quantified interactive and substitutional impact among NPT and PT should serve to support more accurate policy making for state governments to find a better trade-off at an early stage in a pandemic. If cross-state trips are reduced, this would potentially lessen not only new deaths and cases but also new deaths and cases per ICU unit; far less efforts of PTs would be needed to stop the spread of virus; the medical system would operate more smoothly hence the unit efficiency of the medical system would increase.”
“The behavioral responses that we have seen to COVID-19 over the past year, both private and public, have had a powerful impact in ‘flattening the curve,’ reducing peak levels of daily infections and deaths by an order of magnitude relative to predictions of standard epidemiological models. These behavioral responses, however, are forecast to have only a modest impact in reducing the long-term death toll from COVID-19 relative to predictions of standard epidemiological models in the absence of the development of technological solutions such as vaccines or life-saving therapeutics. Absent such technological solutions, the long-run death toll in the United States would approach 1.25 million over a five-year period, even with the private and public efforts at mitigation that have been undertaken.”
“Here in the United States, we have been very fortunate with our success in developing and now implementing effective vaccines against COVID-19. With vaccines, the long-run death toll from COVID-19 is forecast to be roughly 600,000, or about half the level without such a technological solution.”
“In contrast to the case of no technological solutions being developed, strong non-pharmaceutical interventions implemented early on are highly complementary with speedy development of vaccines and life-saving therapeutics in that they save lives by delaying illness and death until such technological solutions are available. This model forecast that plausible additional non-pharmaceutical interventions, applied early on and consistently over time on top of the policies that were implemented at state and local levels, could have reduced the long-term death toll from COVID-19 in the United States to roughly 300,000 over a five-year period…Given the success of a number of countries in containing COVID-19 over the past year while preserving economic activity, it is entirely plausible that such non-pharmaceutical interventions would not have led to high economic costs and, in fact, might have led to better economic outcomes.”

Note the author’s comment that “The onset of pandemic fatigue in late 2020 accounts in large part for the peak in deaths in January 2021,” ie, personal decisions to ease up on NPIs.
