In practice it would look something like "Test every last American [with insurance coverage]" plus the usual assumptions about people without medical insurance not existing.
If it costs tens of billion dollars, so be it. That's far cheaper than shutting the economy for months at time.
This is essentially a war-time effort we are in. I believe government has the authority to mandate that companies with relevant manufacturing capability build and sell these machines in order to combat the virus.
However, sufficient availability of supplies and equipment for this scale of operation is a pretty big if.