To me, car or no car in Sicily is very simple.
In Palermo - no car! (I have no personal experience with Catania, but it seems similar).
For seeing places outside of Palermo, you can have a car or not, as you choose. If you are going between main centers, buses and trains work fine; if you're going to smaller towns or more rural areas, a car is very handy. Once in a city (such as Taormina, Siracusa, etc), you don't use a car to get around.
So, if you are starting in Palermo, don't pick up a car until you leave. If you're ending in Palermo, return the car when you arrive there. If Palermo is in the middle of the trip, try to use public transit to get there (bus or train), or else get very specific directions on where to park. Once you have parked your car in Palermo, you won't touch it until you leave.
For more details of my Sicily trip in 2014, including lots of info about our car rental and driving, here's my trip report: https://community.ricksteves.com/travel-forum/trip-reports/sicily-palermo-caltanisetta-siracusa-taormina
No need to worry about "shady" car rental companies; we used Kemwel, which is a consolidator and our car was through Hertz. The next year, my friend again used Kemwel and his car was through Europcar (not well known in the US, but a major company in Europe). We found Kemwel had, by FAR, the best rate, but do check around.