RT-qPCR is indeed the common detection method right now. RT (reverse transcriptase) converts RNA->DNA, PCR (polymerase chain reaction) doubles DNA by replicating it over and over in heat cool cycles. To a first approximation, you get 2^n copies of the RNA (it was converted to DNA) where n is the number of heat cool cycles. By adding a fluorescent tracer, you get the q (Quantitative), so you can see how much DNA exists at each cycle. You can then fit a curve and see how much RNA (via its DNA complement) existed in the original sample. It's not actually that slow, you can do this in a couple hours if you have the right setup. The bottleneck is you do need time for the heating and cooling cycles. You also need to extract RNA from the sample first, which could be another bottleneck. I'm not totally clear on the relative cost, a good ELISA would eventually be cheaper, but qPCR itself isn't very expensive. The benefit is that qPCR is extremely sensitive, you WILL see if there is virus in the bloodstream. You can even use swabs and avoid blood altogether. Also, the minute you have a viral genome, you can make a qPCR test, you just need the sequence. That said, if you cleared the infection you will be negative under RT-qPCR.

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