Bioinformatician here (well, former): I would never say something is impossible but the way these things are tested right now, it pretty much is impossible.
We need to amplify some molecules and run a process called PCR on them for which you need a full lab setup. It would take nothing short of a revolution in detection to achieve this.
What would the cost be to set up testing machines every few blocks? People just get a swab, drop it into the machine, and get a code (just take a picture of a QR code) that they can later check for results.
We have testing machines which can pump out 900 results/h (in batches of 900). The issue is that nobody is using them for disease testing, they are instead being used for drug discovery by big pharmaceutical companies.
We need to amplify some molecules and run a process called PCR on them for which you need a full lab setup. It would take nothing short of a revolution in detection to achieve this.