IDP: not everywhere in Europe requires it.
Italy, Austria and Spain are the 3 that I believe still do. The rest of Europe may not.

Greece was on that required list until very recently they got rid of it.
It seems outdated and un-necessary to me. It is a very basic booklet AAA provides and is not translated into any foreign language so I don't see how it is any easier to read for a foreigner than any US driver's license and could be MUCH easier forged than any state issued license I have ever had but I suppose is a standardized format.
Good thing is it is inexpensive and really only takes about 15 minutes to get one from a AAA office even if you are not a member so not that much of an inconvenience.
Why AAA has the US monopoly on this I have no idea.

Have never been asked to show one. Have heard stories of rental company offices asking for them and then denying a booking - my theory is in the rare cases this has happened that office had overbooked and uses this as a way out of their problem. Just a guess, it has never happened to me.
Extremely rare to see a police pull a car over for speeding or similar violation as Italy uses speed cameras and will mail tickets rather than have police actively enforcing it on the roads like we have in the US.
Bigger reason to get one would be in case of an accident even if you were not at fault you could be deemed at fault if you don't have an IDP and it also could be seen as a violation of your Insurance agreement with Chase so for that aspect it is worth playing along.

Hopefully in future years it will not be required in Italy.

This post made me think of another tip: if you don't already use the app WAZE download it and use it on your trip in Italy. It works great everywhere in Italy but the real benefit is it will point out where the speed cameras are as well as when within a Tutor zone (measure average speed over a highway portion) so you can speed quite easily on other sections of the highway without fear, lol. Even when I have had GPS in the rental I usually will keep WAZE open in the background for the speed camera aspect as car GPS often will not have that (sometimes they do)