Here's a fun experiment that will never happen. Let's pick two states with roughly the same number of cases relative to population. Keep one on lockdown, open the other up, then measure deaths by all causes and see what we find.

Also, we can measure health and domestic crime rates and diet.

Great idea! Actually we will get some indication with what Sweden is doing. People are saying 'Oh Sweden has a high number of deaths compared to Denmark and Finland' but that's disingenuous coz the no lock down strategy is not meant to be measured as a day-by-day thing - of course there will be more cases in the beginning - there's supposed to be - but it's about who comes out best in the end. That includes damage to the economy and all the repercussions of that.
As you said, measure the total number of deaths from all causes - not just virus deaths - in say one year, two year's time.

Re the data companies getting involved, now it seems Facebook has waded in. I always used to defend FB on the grounds that it provided a platform for many protest groups to organize and co-ordinate, but now they are taking down pages that are trying to organize protests against the lock down.