The last time I drove in Italy, I vowed it would be the last time ever. In fact, I won't drive in Europe. It's usually a royal pain. All the city streets are one way in the wrong direction. Everything is narrow. There's no parking. And you're not allowed in the city center without a special permit which you must buy with your phone, otherwise it's a hefty fine.
I'll never do it again.
If the timeshare isn't within walking distance of a rail line, I won't bother. That knocks more than half the resorts out of contention -- because they're out in the country, away from train stations. They're drive-to destinations for vacationing Europeans. (And why wouldn't they build it that way? Europeans are better at vacationing than Americans are.)
There are only a handful of timeshares in Rome. And the chances of exchanging into one are very slim. If this were my trip, I would fly into Fiumicino and stay in a hotel/AirBnB for several days. And then I would stay a week elsewhere in a timeshare on a rail line -- pretty much anywhere in the country. Other than Mestre, I've never been anywhere in Italy and said, "Wow, this sucks." (And lots of people stay there because it's the closest city to Venice.)
You can see an impressive amount using your timeshare as the hub. And then fly home using whatever airport is most logical. (It's almost always Malpensa.)
My early timesharing in Europe was designed around using a timeshare as a base to explore the area, but I often found it more efficient to book hotels and drive point to point. A few years ago, we did two week-long trips to Poland, one to Silesia, and the other to Prussia / Pomerania and stayed every night in a different castle or palace, all booked through Booking.com and costing $40-$60 per night. The final castle we stayed in had been the retirement home of Prussian Field Marshall Blucher who commanded Prussian forces in the Napoleonic Wars and had been bought from the Polish government after the fall of communism and restored as a hotel by a member of the Blucher family.
After a time, unless I had an exchange into an urban area where I wanted to spend a week, like Venice (through DAE), London (through SFX), and Vienna (through RCI), or a beach timeshare like the Riviera, Greek islands, Malta, Croatian coast, and Canary Islands , I only used a rural timeshare to travel point to point if the timeshare itself had something of an appeal to it, like being housed in a historic building, like several in manor houses in the UK, a couple of chateaux in France, and medieval Austrian castle Schloss Grubhof (which is no longer a timeshare).
Driving in cities can be a pain in the US or Europe, and NYC or Boston can be as challenging as major European cities. I have found Greek drivers the worst, and was rear ended by one on the island of Corfu but I have driven there twice and would do so again. Malta has a reputation for bad drivers but they seemed much tamer than Greece. I have driven in the UK, Ireland, France, Germany, Austria, Switzerland, Luxemburg, Belgium, the Netherlands, Monaco, Liechtenstein, Denmark, Andorra, Spain, the Canary Islands, Malta, Poland, Lithuania, the Czech Republic, Slovakia, Slovenia, Croatia, Montenegro, Bosnia, Greece, Bulgaria, Romania, Ukraine, Hungary, and Moldova. There are few places I would avoid driving, but if I wanted to spend several days in a big city, I would put that at one end of a trip and not keep a rental car during that time. I tried keeping a rental car in Prague once after a trip around the countryside, and decided that had been a bad idea. Parking was terrible and it was easier to get around on public transportation.
One thing to remember about car rentals in Europe is that most Europeans drive stick shifts, not automatics, and you will generally pay a good bit more to rent a car with automatic transmission. I like driving a stick shift, so that has never bothered me, but I know a lot of Americans who have never driven one.