I would be willing to bet that if you combine all the different lists that New York State and its various agencies have (DMV, DOE, Department of Taxation and Finance, NYC ID, jury duty, voter registration, social services, etc.), that you would easily get >99% coverage of all people who've resided here for at least one year.

This would be a much better list to sample randomly from than "go to a grocery store and test everyone who walks in".

I should point out on /r/nyc, some local redditors saw the testing going on all week in the same location and posted about it, informing others. I suspect this led people who wanted a free test to actively seek them out, especially because it's so hard to get tested otherwise. I'm pretty sure I had it over a month ago and I still haven't gotten tested, so if I'd seen those posts in time I'd have headed over there to get tested myself. Point is, the sample is even further biased because word spread around and some number of people getting tested there were actively seeking it out for reasons.

Now you're talking about a huge legal issue just to get access to the data, followed by a huge record linkage issue to remove duplicates. So with the time pressure involved, your proposal is so completely impractical as to be ridiculous.

These are all state agencies. They're already sharing data with each other anyway (e.g. the jury selection tool is getting feeds from many of these other sources).

What huge legal issues? This is all the government. Of course it has lists of all of its citizens, and can and does use said lists.