There's one general point that has to always be considered in these sort of ideas. Resources are not infinite. Of course you realize this, but I think we who come from the software world can fail to appreciate how severely restricting a problem this actually is. There are 325 million people in the US. Let's imagine a very high participation rate and just exclude infants and some outliers, just so see what would happen. So we'll call it 300 million.
That's 300 million tests per day or about 110 billion per year. The cost there is already going to be extreme. Even if you can get the tests down to $10/person, which you probably cannot, that's more than a trillion dollars a year -- upwards of 5% of the GDP. But the bigger issue here is what money represents: resources.
Machinery, maintenance, operation, even the strain on the transportation system would be intolerable. For instance think about the little kit you send over with your sample. I mean you'd think just getting it there would be negligible, but it'd be anything but. The USPS currently handles 20.2 million pieces of mail per hour. [1] Just handling these packages alone would increase the entire postal system's burden by more than 60%!
Or even consider the simple packaging that needs to be produced and disposed of. How small of a kit can we get to ensure sterility, sufficient sample size, and transportation safety? Let's say 100 grams. That doesn't seem like that much, but again think about it at scale. That's 30 million kilograms. For some sort of scale, The Statue of Liberty weighs about 200,000 kilograms. So each day you're talking about 150 Statues of Liberty of packaging and material that needs to be manufactured, distributed, delivered, processed, and disposed.
Big ideas in the physical world have quite tremendous barriers.
---
As one interesting aside, imagine you took the market value for literally every single consumed good and service in the US, the sum total of money invested by businesses, and all government expenditure on top of it. And then you divided that equally among every man, woman, and child. How much would it be? It's interesting to compare expectation versus reality there. Conveniently what I described is really easy to measure - it's the GDP/capita. And it's less than $60,000. I always find that number quite bemusing when thinking about it contrasted against our economic ideals in various matters.
[1] - https://facts.usps.com/one-day/