There's yet another reason: a test can provide only a certain amount of information gain. If the probability to have contracted Covid-19 before the test is too low, even though we know more after the test we may not be able to draw practical conclusions that are any different from the recommendations without test. In other words, if the recommendation goes from "stay at home and avoid contact to others" to "stay at home and avoid contact to others" the test would be wasted.
The Covid-19 tests are not suitable for screening purposes (i.e. searching for infections among a population where infections are [still] very rare). They are to sort out who is infected and who isn't among high-risk populations where 1 in 10 (or more) are actually infected.
You may want to look at my longish answer to How accurate are coronavirus tests? for relevant background.
E.g., with the newly emergency approved Roche test, we may say that a positive test result increases the odds of having Covid-19 by factor of somewhere around 30 - 100, a negative result decreases the odds by a factor of 1/50 - 1/17.
If you are in a risk group with a prevalence of 8 %, the pre-test odds of 8 : 92
- increase to (240 to 800) : 92 or a post-test probability of having Covid-19 of 70 to 90 % with a positive test result, and
- decrease to 8 : (4600 to 1564), i.e. the post-test probability of having Covid-19 is somewhere around 0.5 to 0.2 %
These post-test probabilities allow practical conclusions: if you're negative, you're fine and can be let to meet the public, if you're positive you need to go to quarantine and/or treatment.
8 % prevalence is my guesstimate for those who are currently (Mar 15th) tested in the US (and incidentally, also in Germany).
Now consider the overall US population. With currently 1629 cases and a population of 328 mio., we'd get pre-test odds of 1629 : 328 mio*.
The test results change this to
- positive result: (27700 to 162900) : 328 mio or a post-test probability of having Covid-19 of 0.008 to 0.005 %
- negative result: we don't even need to calculate this, because the pre-test probability was only 0.0005 %, post-test probability is lower.
Even if we assume that there's a dark number of yet unknown Covid-19 cases, say, a factor 20 over the known cases, the post-test probabiliy after a positive test is about 0.2 % to 1%.
What would be the practical conclusion? Well, probably that the positive case should stay at home as much as possible and avoid contacts. Certainly not that they can rely on acquiring immunity against SARS-CoV-2.
Even for low risk groups who have flu-like symptoms but no known contact to Covid infected people (or other high-risk situations) the pre-test probability of having Covid is too low to allow the test to make a meaningful difference: after all, even if it is "only a common flu", they should stay at home, get well and not infect others meanwhile.
While the pre-test probability is likely to increase (because more people get infected), in order to keep the health system/hospitals working the recommendation will probably not change: stay at home in self-quarantine as long as the symptoms are mild, so that the health system is not burdened by cases that get well on their own.
We may also say that were the whole US population to be tested in that situation, the 32500 true cases of Covid would be drowned in 3 - 10 mio. false positve cases.
*328 mio - 1629 ≈ 328 mio