I know that qPCR is preferred for diagnostics, but is it actually necessary? As in, will the false positive/false negative rates suffer enough for standard RT-PCR to be useless?
No. qPCR is, well, quantitative, so it gives more information. Additionally, it requires less pipetting, and the data is easier to report. All of these things make it more desirable for diagnostics.
No you just need more of those parts, and depending on how much manual labor you want to be doing a different number and configuration.
process is:
-take sample
-amplify via thermocycler
-validate presence
That last step can be done a few different ways depending on what you are looking for and how much time/money you have. (I'm just a hobbyist so not getting into specifics to avoid misspeaking but I'm confident it's possible via open tech for the 'crude' methods -- more advanced are more proprietary i'm sure.)