Thanks. I've only had a chance to have a very quick glance at the paper, but my first reaction is that one has to regard their analysis of lockdowns with great caution, since they have treated 'lockdowns' so simplistically.Here's a scatter of lockdown stringency vs outcome for a range of countries. Source: clicky .... The authors found some factors, age, health etc produced some correlation with morbidity.
As you say, they have examined the relationship between 'lockdown stringency index' (basically a measure of the strictness/severity of the lockdown) and mortality (not morbidity as you wrote) in many countries. However, as we know only too well from what many people think about the UK's experience, 'stringency' is but one characteristic of something quite complex - to simply reduce a lockdown to a single number, and then plot that on a scatter plot and feed it into regression analyses is a great over-simplification, which I would feel is verging on the naïve. For a start, the timing of a lockdown is probably even more important than its 'stringency'.
As you will be aware, one of the greatest criticisms of the government's management of the initial wave of infection is that, on the basis of retrospective modelling, some believe that had the March 2020 lockdown started just a couple of weeks earlier than it did (which is apparently what some of their advisors were advising), there would probably have been "tens of thousands" less deaths, and some believe that even more tens of thousands of deaths (in Jan/Feb 2021) would have been avoided had our lockdown which started in early November 2020 not been 'interrupted' for most of December. Hence, just a couple of 'political decisions' could probably have dramatically altered where the UK appears on that scatterplot - and the same many be true of many countries.
More generally, as we knows, one has to be very cautious about any attempts to compare countries. Although, as you imply, these folk attempted to look at quite a few factors which could differ between countries (and hence affect morbidity, with or without lockdowns), there are undoubtedly many (maybe 'unknown') factors that they will not have been able to consider. Indeed, although 'mortality' sounds like a pretty hard metric to model, there is very considerable variation between countries (not to mention within the UK!) as to how a "Covid-19 Death" is defined - so one cannot even be certain that the 'deaths' figures are necessarily comparable between countries.
Probably more when I've had a chance to look at the paper properly.
Kind Regards, John