It does to an extent. You can't just blanket say well then I'll repeat the test. If the sensitivity is only 70%, you could end up with 2 false negatives in 1/10 people who then will go out and spread a highly contagious virus none the wiser (and given we had 300+ positives here again, you can do the math on how that works), which completely negates the point of what you're trying to do. There needs to be a minimum threshold which is why even here with the current tests we are double ruling out people who get admitted if they're high suspicion because the test sensitivity isn't great. This costs time, ppe, beds, and staff to do all of which add up. In a world where there is no money and there is unlimited reagent and lab capacity, sure test til the cows come home and have no idea what to do with your results.
On the point of wanting mandated reporting, it is obviously important to know the case rate and infection penetrance so you can make educated decisions. Like on things like reopening business, schools, sports, etc. I agree that in home testing would be great and would really pull a large burden off the system, but if it's not going to be reported nor able to be enforced nor able to be acted on, it ends up causing even more confusion.
I understand what you're saying and yes the organization should do everything it can to be nimble in this endeavor (and if they are acting in ways that only seek to make the president look bad, they're pathetic human beings, which hey I guess could be the csae), but it also needs to come with the understanding that you can't just churn out bad science and hope it sticks. It's why we've had an array of fucking chaos to this point because people have fucked the data from day one.