In my opinion. Anything you will be doing touristy as well as transit is relatively safe and no issue. In the many talks I had with Jordanians they made it seem like their main hatred is Israeli not as much Jewish. I said I was Jewish and they always asked "from Israel?"
The tourism economy in Jordan barely exists.
I went to the citadel and there were a few Asians.
I went to the Royal car museum and there were a couple other people there.
I walked around rainbow street and a souk and maybe saw 10 tourists.
These are the only things for tourists to do in Amman.
This was a few weeks ago. Maybe there are more popular times?
As far as safety goes, I was never nervous.
Though I still wouldn't recommend leaving the airport if your a family or you look Jewish.
Though I don't see any issue transiting AMM without leaving the airport.