Self-administered = you're asking people who have no idea how to do it right (and are possibly ill) perform it on themselves. That's unprecedented in medicine. Glucose testing? Colo guard? In a similar vein, these are screening tests. Hardly unprecedented. Further tests would be needed by a professional.
Hardly invasive? => You are jamming a stick well up into the membranes of your nose, while trusting that the stick is sterile The swab is opened in front of you. If you don’t trust that, I’ll make you a tin foil hat. Manufacturing/processing errors might occur, but they’re extremely rare. Your sinuses are a filtering zone to your body, anyway. Minimal risk.
Minimal (to zero) risk of exposure => That's also true of riding alone in your car, and yet daily I see people doing that while wearing a mask. And college students are being told they have to wear a mask while walking alone on campus. So it's ok to advocate, demonstrate, and enforce senseless policies wrt wearing a mask to avoid minimal chance of exposure, but just throw that all out the window for the testing? Whataboutism for silly people and their behavior (who might be carrying a Doordash or driving for Lyft/Uber) or policies that might be a little bizarre doesn’t change the risk for the testing.