Direct detection of viral particles is substantially harder because for N viral particles you basically get O(N) signal (to put it in CS terms). There are some caveats, your readout antibody can generate an exponential signal for example.
You still need to be presenting the right part of the capsid and have the right material. In the direct ELISA scenario, you need 2 antibodies. One to bind the capsid to your surface, the other to bind it again and either have a readout attached directly or be bound by yet another antibody with the readout (likely the latter). Fortunately, viral capsids are repetitive so you might be able to use the same antibody for both. You ALSO need those antibodies to have low cross-reactivity with sputum or whatever other rough sample you have. When you are looking for a patient's antibodies, the patient's immune system has already taken care of that for you. This is all tricky, obviously it's not THAT hard, but it's not trivial.
RT-qPCR is directly a O(2^N) signal. It's extremely sensitive, requires very little assay development, can be made very specific, can work with little sample prep, so it's a much better front-line, first-to-go test than direct ELISA. Still, we'd all love an ELISA, particularly to patient antibodies.
The signal is actually more like N*2^C where C is the number of cycles (usually around 30). So a huge amplification, but still linear in the starting amount, i.e. O(N). That's actually more useful, generally, since ratios of amplified material stay approximately constant during amplification.
The false negative reported several times from the same patient is interesting, but may be more due to testing source (upper respiratory fluid vs. sputum or lower respiratory sources or blood).
(that is, what's the scale of the fuck up that the machine testing is still being brought online)
From what I can tell (mostly anecdotal and other data from friends I trust) the CDC assay used bad primers, they were noisy, showing a bunch of false positives. They also used a potentially dodgy fluorescence readout. I used to use the same readout, but our scenario was full of internal controls and the input sample was far more consistent. For human sample data a different fluorescent method that is more reliable should have been used.
Important ways this can be screwed up: it's helpful to use the same machine, the same protocol every time. This is because qPCR is an exponential assay, which means small errors can have big repercussions.
TLDR: Scale of fuckup here is massive. This really isn't that hard, and we could just have used what the WHO/others were using. I have no idea why they decided to go their own way.
more info: https://www.propublica.org/article/cdc-coronavirus-covid-19-...
An ELISA works differently. In this case you would present a part of the viral protein, the patient's antibodies will bind to it, and you then use another antibody (this one is conjugated to some sort of readout method, fluorescent or biochemical) which binds to generic human antibody. Now you have a stack: viral protein <> patient antibody <> readout antibody. You read out the signal provided by your readout, usually some sort of colorimetric thing (see home pregnancy tests, also an ELISA, slightly different configuration though). This works fast, can be manufactured in bulk, BUT requires do you have a protein(s) (in this case the viral protein) that is/are universally recognized by patient antibodies. It's a little trickier to develop. You also need to produce that protein at scale which can take a little time as well. Major benefit: with a good ELISA you WILL see not only whether someone IS infected but whether someone WAS infected for some period after illness regardless of symptoms. This is essentially the only way you'll get a really good number for baseline infection rate. That said, you likely actually need a blood draw.
In short, we need both, we probably could do a lot better than we are doing with qPCR in testing volumes and we could also really use an ELISA.
Source: PhD biochemist, have personally run these assays in various forms.
EDIT: Acronym expansion, source