Long, potentially boring post ahead - on the topic of federal labs.
As to why Federal Labs aren't much help in a pandemic (and the related question of why the military isn't able to perform testing), I can speak to the ones in NM, SD and WY (but the data on the others is similar).
These are not medical or biological labs. Coronavirus is made of RNA, and needs a machine called a PCR to detect it in a nasal or cheek swab.
The federal labs study things like major toxins, test military and civilian weapons (particularly components nuclear weapons, but also the effects of tear gas and other toxins). They study satellite imagery to try and detect banned nuclear activity around the globe, oversee radioactivity detection at the nuclear labs and in the nuclear weapons production sector, etc.
They use equipment like mass spectrometry to analyze all kinds of toxins and also, unusual elements of machinery found in military contexts. That's not a machine for studying RNA.
To train military personnel to work with PCR machines is quite the challenge. I teach a very low level class that eventually leads to the kind of training one needs to work in an actual lab and I would say that military students, while hard working and dedicated, are no more talented at understanding the complexities of DNA/RNA than anyone else - and it takes about 3 years to get a beginning position collecting biomaterials (keep in mind that the swabs themselves are toxic and sending through the mail would be a big problem).
The labs need to be near the subjects. I do believe AZ has used labs just outside its state, but there are big questions about how long swabs can be kept without analysis (it's more than 2 weeks for many Arizonans and it's the same elsewhere). If the swabs were sent by mail or by military transfer, the chain of custody would need to be very carefully set up (and that would be only if the federal labs would switch to being genetic/PCR labs - which they can't easily do, because their other research is too important).
PCR machines are cheap and the existing hospital and medical lab doctors and techs know how to use them - but we lack additional techs in the labs. Quest Diagnostics in California purchased more machines; every large hospital has some; many teaching hospitals and universities in California converted their PCR machines to process way more samples at a time - and that's what we need. We do not lack physical labs, but we would need a massive movement to get the technology for faster processing of tests out to every other state - and AFAIK, that's just not happening. The federal government could coordinate that.
But to have military or postal service doing "special delivery" of dangerous biological hazard material over state lines and long distances might require refrigerator trucks and specially trained personnel. It would be slow. Military airplanes would have to be specially commissioned for this and in many states, a military convoy would then be needed - all day, every day.
Can people run PCR machines, taught by rote, without any knowledge of how the machines are supposed to work? I don't think so, personally. We have reference labs, checking through various measures, to make sure other lab personnel are performing properly. That system would need to include federal labs too - and just getting certified as a biomedical lab takes time (and no, I would not want to see *any* steps skipped in this pandemic).
The military does have PCR machines and labs to do this work for its own personnel - but I'm not impressed with the speed and thoroughness with which they are doing the work to begin with.
Sorry for the long post, but my fieldtrips to Los Alamos, Sandia and other places were eye-opening. Our neighbor (Airforce) traveled the world inspecting our military's use of nuclear components and brought the samples back to NM for analysis. The labs I visited were almost exclusively devoted to testing various components of nuclear weapons, nuclear reactors, etc. Today, Los Alamos is also a major study site for the various mutations of CoVid, including analysis of the spike proteins that are wreaking so much havock in their bodies. We need them to do this work, not to have our nation's top scientists involved in trying to process nasal swabs from around the country (joining and training in the CIS side alone would be an enormous disruption to the national level work those labs do).
Los Alamos National Lab: National Security Science
These labs are also looking a tiny aspects of the human body and the human genome that relate to CoVid (tiny in size, not impact). Their PCR machines are hard at work trying to understand the walls of the human heart, for example. But there's no way that the average person in the military could walk in and start operating the machinery or calibrating machines daily. And I doubt that the labs want any computer bridges built that would transmit results to places like Quest, so that patients could receive them (another area in which most states are very slow - results can be known, but the patients don't know, because the CIS isn't working to capacity).
Lots to fix - but I don't see a large role for federal labs or the military. University hospitals can create very large capacity for testing - if they are funded.