The degree to which there is compliance with an intervention is an integral part of most major research studies, and many smaller ones. Even if there are people violating the lockdown, it can still be highly effective and be seen to be such in the data. If you are willing to look for it, which the Cato economist and his buddies were not.You forgot to mention something obvious. Any study that tries to measure whether lockdowns worked or didn't, especially in the US, is BS. How do you account for the fact that a huge # of people ignored the mandate and did whatever the hell they wanted? There is no data on this, so any analysis is fatally compromised.
It seems that the situation now is simply one of how to mitigate the strain on hospitals while Omicron comes through. Masks would certainly help. Vaccinations would help infinitely more.