US testing started out slow but it's now up to 100,000 tests per day (and more than half a million total), which I think is the most of any country (though probably not per capita)
No, they're testing people who are a) having issues as likely infections or b) those who have been exposed.
The odds of any random person getting it are still low, especially if you're not in one of the epicenters. Testing random people would be wasting time, effort, and tests. All of which are limited.
I'm interested in the theory that a fair amount of untested people have or had it, and are mostly asymptomatic. Iceland did some work there: https://www.buzzfeed.com/albertonardelli/coronavirus-testing... (Not my favorite source, but plenty of direct quotes)
That's a safe assumption since ~80% are asymptomatic and therefore probably think they don't have it or it's just a cold, allergies, etc.
That's where the "stay home!" mantra kicks in. If you have it and stay home for two weeks, maybe you infect your family but none of you infect anyone else. Therefore, it burns out without anyone else to spread to. Not a perfect world but it's the one we ahve.
As far as I'm aware, no country is doing that yet except Singapore; but fifteen minutes ago they announced that they would be supporting serological studies to reconcile the models with the numbers. You could think of the numbers we're seeing as a diagonal 2D slice on a more complex manifold: we're either seeing spread like this because there is a large mass of asymptomatic carriers, or because there are unexpectedly-tenacious mechanisms for infection.
A serological study, which you would usually only conduct post-mortem (but in light of the conditions, we may have use for an interim study), would shed light on how many people had been infected without knowledge, and give us some estimates of how infectious asymptomatic carriers are.
Short of that, in countries where testing has been highly available and unconditional, you can begin to get a picture of that; it's not as good though, since people who go for testing are still self-selecting to some extent.
Source? I can't find any information that Singapore's testing is random.
Also, if it is I stand corrected on my original statement. However, Singapore is a pretty small, cohesive nation with a strong central government. They, along with a handful of other city-states, are uniquely positioned for this. The poster child for testing-based control, South Korea, is not testing randomly. So, how is this a dig on the US?
https://covidtracking.com/us-daily/
https://ourworldindata.org/coronavirus-testing-source-data (hasn't been updated in a week)