The int's driver license is more for your protection in case you had an accident that required police intervention. Police has the legal right to ask you to show the international driver license.
Rental car companies in Italy generally dont care. We never encounter a single rental car company asking us about the intl driver license when picking up our rentals, whether in Sicily 2018 or in Puglia (Brindisi) 2019 or in Rome 2023. All were rented from US companies though.
