Has this been proven in any country?
If I were to guess, I'd guess China. But in the actual numbers, the DALYs lost won't be measurable for a long, long time. They went through aggressive lockdowns plus aggressive vaccinations and (I think) had a pretty low covid-19 case count, which means that the lockdowns could easily have caused more damage.
Remember, a
successful containment could easily cause more lockdown deaths than covid-19 deaths. When it comes to a pandemic, a successful response looks like an over-reaction in retrospect, because you have to err on the side of being too aggressive. Now, usually I bring this up when talking about money, since you need to flood in resources quickly and then 'success' looks like you spent too much money per individual case. Of course, under-investing means that you spend way more money in the longrun. By analogy, it's always possible to have doused your campfire with less water, so technically there was 'wastage'.
Theoretically, the way to maximize lockdown deaths is to never contain the virus, but also never let it explode. That way, the grind of the lockdowns does all the damage. The alternative, where you just don't lock down
quite sufficiently means that you can really let Covid-19 roll through and cause the most damage, but if you can drag that out as well then the lockdown deaths mount. This tends to capture Inno's concern, where we just crank up both sets of damage in perpetuity instead of biting the lockdown bullet harder than we want to.
Once an epidemic gets to a high enough penetration of the society, distinguishing lockdown deaths from pandemic deaths gets very hard. My province's ICU were empty last summer, aggressively so, and so other essential health services weren't being provided. But this spring, those ICUs were full of Covid-19 patients. While a quick snapshot of distinguishing the first as 'lockdown caused' and the 2nd as 'pandemic caused' isn't so hard, it's still hard calculating the damage. Every heart attack patient that failed to get properly treated last summer will show the same 'cause of death' as the heart attack patient that couldn't get treated this spring.