A solution that works outside of the vicinity of New York, would still allow at least 47 other states to open up.
As for checking who has a test, simply give colored stickers. If someone wants to “beat” the system, so be it. Social disapproval and common sense will keep most people honest.
New York City isn't the only place in which car ownership is low. What about Chicago, Philadelphia, San Francisco, Los Angeles, Boston, DC, etc? If your plan for reopening the nation doesn't include reopening our cities, it isn't a real plan for reopening the nation.
You need a plan for people who want to beat the system because they present a huge danger. The whole idea behind this system is to allow the people inside the secured bubble to return to normal behavior. They aren't going to be wearing masks, social distancing, or taking other precautionary measures. Therefore one person acting inappropriately could present a huge problem for the people on the inside. Keep in mind there are still people who think this entire thing is a conspiracy and that COVID-19 is no worse than the flu. You have to consider what happens with people like that who might not participate in this system in good faith.
I wouldn’t plan on reopening the nation, since it was never closed from the top down to begin with. Individual states, and in many cases cities and counties, made the decision and continue even now to enforce rules different from one another.
Hawaii is an island, thousands of miles away from the rest of the USA, so why shouldn’t it open on a different schedule?
Even China, ground zero for the crisis, close and reopened different providences, districts, and even neighborhoods independently.
As for checking who has a test, simply give colored stickers. If someone wants to “beat” the system, so be it. Social disapproval and common sense will keep most people honest.