There's no stopping anyone from getting the primers (small pieces of DNA) and doing a qPCR yourself. You'll also need reverse transcriptase (the RT, converts RNA->DNA) and a way to purify RNA from a mix of cells/spit/junk. These are all relatively inexpensive, standard molecular biology reagents. There are some shortages of RNA extraction kits, but a good molecular biologist can do without one.

What you might be missing is consistency, which is what the medical system is worried about. You want to avoid false negatives AND false positives when it really counts. Getting that consistency is where the regulations come in. Nowadays I think it's appropriate to relax them a little in the interest of the emergency.