I haven't seen any recent messages about what the experience is like flying into Palermo as of late August, so I thought I'd share my experience for those of you who may be considering taking the trip.
We arrived in Sicily via PMO on August 28, on a United flight originating in the US, and transiting through Switzerland. When checking-in for the flight we were asked to show our vaccination cards, the dPLFs for Italy, and a similar Swiss PLF for Switzerland.
As we exited the baggage pickup area in Palermo, we were asked where we had traveled from, and when we answered "United States", we were told we needed to take a COVID test, and that we should follow the signs to the "COVID info" area. We mentioned that we were vaccinated and had already taken a COVID test just before we left, but he said it didn't matter, everyone from the US had to take a COVID test. (As an aside, I'm not sure if or how this was enforced. There did not seem to be anyone monitoring us, and it seemed like we could have just ignored him and gone to pick up our rental car).
We followed orange "COVID info" signs to the middle of the arrivals area, where there was a booth with two more gentlemen answering questions (who fortunately spoke English). There was a QR code that we were directed to scan with our phones, which linked to a website with a form to fill out (which was a somewhat simplified version of a form we found before we left, but weren't quite sure how to fill out). After filling out the form, the website generated a code, which we were told to take to the testing area. At this point we exited the airport, and followed more orange signs to another building across a parking lot. It took probably about 10 minutes of walking, which was a bit unpleasant when lugging all our baggage in the warm evening sun.
The next building seemed quite disorganized, with a crowd of people all trying to get in. When we eventually got near the front, we were asked where we were from, and then asked to show our vaccination cards. We were then allowed into the building and formed another line. After a 5 to 10 minute wait we reached the front of this line, where we showed the code from the website, and were given a numbered ticket. After waiting another 5 to 10 minutes we reached the test area, where we got the full-on tickle-the-brain swab in both nostrils. Then into a waiting room where we waited for another 10 minutes for our results. We weren't told how the results would be communicated, but after 10 minutes realized that it arrived through an email to the address we had entered in the form we filled out (as an attachment, all in Italian, with the result "NEGATIVO" at the end). We showed this to the person (guard?) at the exit door of the room, and then were allowed to leave. Another 10 minute hike back to the airport terminal (lugging all our bags), and we were finally able to pick our rental car and head to the hotel.
So, for anyone planning to go through Palermo in the near future, make sure to budget some extra time for the COVID test. Total time spent at the airport from arrival to leaving in the rental car was about 2 hours, and I think about half that time was COVID test process.