I have to say, I think SA is being far more sensible over their borders. I'm planning my travels for next week. I apply for a permit to cross the NSW border; on-line application, nothing required to be uploaded, e.mail came through within 90 seconds, granting me permission to cross from Victoria to NSW. Very user friendly but possibly explains why out of 30,000 vehicles, only 12 have been turned back.
In contrast, I apply on-line for SA. I fill out a form, not dissimilar to NSW, but it has space to explain exactly why I wish to cross the border, it requires me to upload supporting documents and then I get a nice e.mail from SAPOL, telling me they'll let me know in 72 hours, but if I'm in a hurry, take my supporting documents and try my luck at the border.
Having gone down the "try my luck" path previously, it wasn't quick but it was very professional. My paper-work was good, forms were filled out and I was sent on my way. I like that SA have provided an on-line way to smooth the path prior to setting out, but if you're in a hurry, you are still able to make your case at the border.