Late reply, but yes to the former, with the caveat that it's contingent on sensitivity and specificity. (I.e. false negatives/positives)
Probably yes to the latter, unless it takes away resources we need more urgently elsewhere.
But really, what we need to do is depress the count of infected people to a point where it's even feasible to test everybody with symptoms.
Even if we rolled out testing to a point where we can test 1M/day (I think it's 200K/day right now?), it'd take a year to test all of the US. If we're in flu-prevalence territory, that's only 19M, and so suddenly much more doable.
And at that level, we need the ability to both test anybody with symptoms and to conduct random samplings of asymptomatic people, because without that, we couldn't lift shelter-in-place until we have a vaccine. That's... a tad long.
So, more tests, plus suppression doing its job. And it'd be lovely if our government would push to get anything done, really. They've wasted 8 weeks by now, at least. Coordinated response is allowed to kick in any day, by my book.