The swab absolutely is not safe.
There are a myriad of reports of different swab products out there that say that many of the swabs don't just contain harmless cotton - it's a mix of glass fibers, some nano-powder and various other materials that get stuck in that sensitive area.
However there is absolutely a way around it. Just say that you have a deviated septum or some other nasal condition and simply won't take the nasal swab.
Mouth is fine. I have multiple friends and did the mouth swabs myself. You just remain adamant and most testers allow it to be done. And don't do the tests at the airports with your flight being due in 2 hours. Because then you may not have the option of arguing with this one new anal TSA-Covid-agent who knows that your flight is about to take off and you don't have enough time to ask for his boss or find another place.
And even in the mouth - rinsing your mouth is good advice afterwards.
The tests however do make people sick - especially the ones performed in the nose. The virus is so dangerous that a sneeze in a bus transmits it to 20 people, but they need to take the sample from your brain - mouth should be enough. You can use such arguments with obstructive workers.
I searched deviated septum and I was surprised to realize that I actually have it. Not very severe but enough for it to not be a lie.However there is absolutely a way around it. Just say that you have a deviated septum or some other nasal condition and simply won't take the nasal swab.