The problem with wide scale testing is that it makes the CFR incomparable to seasonal flu or previous outbreaks. CFR is by definition deaths/confirmed cases. It's
not a measure of infection lethality or mortality rate, even though that's what most of us think it is and what we're most curious about.
For crisis management and planning, you want a CFR that is comparable to something like the seasonal flu or previous outbreaks. "Confirmed" cases of respiratory illnesses are roughly equivalent to the number of people who present to a hospital with severe respiratory symptoms, minus the base rate. We don't go out and run an assay for any person suspected of having the flu, for example, especially not asymptomatic people. If you do that for SARS-CoV-2, the number of "confirmed" cases will sky rocket and the CFR will drop. But that lower CFR can't be meaningfully compared to anything else, and is pretty much useless except as a curiosity.
So it's understandable that the CDC isn't keen on widespread testing; it'll ruin the data, or at least ruin the characteristics of the data most important for crisis management. What matters now from a public health perspective is knowing whether SARS-CoV-2 is circulating. If you know it's circulating in an area, then you should just assume any respiratory illness is a SARS-CoV-2 and quarantine. Test if and only if it becomes severe and you need to know the cause for treatment.[1] You can calculate a more accurate and consistent CFR later by subtracting the base rate of presented respiratory infections from the outbreak rate.
Identifying a local outbreak does request testing, but the major problem there was a temporary problem with getting assay kits out. But going forward, it doesn't help to test people who aren't presenting with severe symptoms. At least, it doesn't help from an epidemiological perspective.
[1] However, SARS-CoV-2 causes primary viral pneumonia. By contrast, severe symptoms from seasonal flu are caused by secondary, bacterial pneumonia. There's not much that can be done for the former; for the latter you can prescribe antibiotics. So for severe cases it might just make sense to prescribe antibiotics, which would be benign for the viral infection (COVID-19) and even possibly a prophylactic for secondary bacterial pneumonia. In other words, in terms of treating a severely ill patient with a clear case of pneumonia, positive or negative COVID-19 identification might not provide any benefit.