Driving in Italy

If that is your whole itinerary, it's not even close - take the train. The van would be a stressful burden.

You say "flying into and out of Rome, and traveling to Florence and Milan". You do know you cannot drive around Rome, Florence or Milan? You have to park outside the centres (pay) and walk or get public transport in.

The decider of me would be where exactly you want to go. Small rural towns and villages a car is useful to get to them, though in many cases the centre is banned to cars and you still have to take paid parking and walk from there.
In big cities a car is a useless liability.

P.S- Never assume a hotel has parking, always ask.

Is the person who wants to rent a van also volunteering to do all the driving?

If you are going to go directly to these cities a car/van would be a hindrance. While in the cites there would be no/expensive parking and you would have to combat the ZTLs. Far a group of 5 you may save some money but not much especially if you make your train reservations months ahead.

A car/van would be a good idea only if you are planning to take days between these cities to see the smaller towns.

That is not a great idea....large vehicles do not get around well in Italy and as much as I enjoy driving there I would not attempt Rome or Florence...and I am still traumatized by a cab ride in Milano...OMG...we were on the trolley tracks and there was one coming right at us. The cabbie was cursing the traffic he was passing because they would not let him back in...too vivid. Yes, I can repeat some of it...I learned a new cadence.
Take the trains. Rent a car if you want to explore Tuscany, Umbria..don't take a car into those cities.

We need more details.

I have driven throughout Tuscany, picking up and dropping off the car at the Rome FCO airport.

The pros:

• Flexibility. You're not locked into a particular schedule to get from point A to point B. Also, if you want to change plans on a whim, it's easy to do.
• Accessibility. There are many areas that are hard to reach with public transportation. Buses take the main roads. We had a blast traveling on country roads far off the beaten trail.

The Cons:

• As mentioned, a car is an expensive burden in large cities. We stayed at an agriturismo in the countryside and took the bus into Florence. We also returned our car in Rome and then took a shuttle bus into the city. We did the same on our trip to Scotland, taking commuter trains into Glasgow and Edinburgh.
• Tickets. We had a great time and came home ticket free. However, I did extensive research on driving in Italy before I left, including signage. I credit my success on obeying speed limits and caution when driving into towns. I never, ever exceeded the speed limit. Out in the country, there were times the locals wanted to go faster. I found the nearest place to turnout safely and let them by. I waved as they passed and usually got a friendly wave of appreciation in return. Personally, I think ZTLs are easy to avoid outside large cities. When in Florence on the bus and walking around, I did notice plenty of signage for ZTLs. Not having to deal with it was welcome.
• Winery touring. DWI is strictly enforced and even a two glasses of wine at lunch will put you at or over the limit. Unless you have a driver who is a teetotaler anyway, this might be a consideration. There are plenty of wine tours that allow you to taste responsibly.
• Vehicle size. I would not want to try to negotiate in a large van. If there are 6 of you, I'd would rather do two cars instead.

Have a great time! There should be plenty of harvest festivals that time of year. They would be well worth visiting.

If you plan to visit just those 3 cities then the only logical thing to do is connect them by train. You can't drive to sights in the cities - there's no driving from your hotel and parking at the Colosseum, for example. You would need to park the van in a garage outside the city, then go to your hotel, and walk or take transit/taxis to the sights in the city. If you found a hotel with parking it would likely be far away from the city center, necessitating taking long cab or bus rides into the city, which doesn't make a lot of sense as you would be limiting the time seeing what you came to see - and it makes it very hard to enjoy early mornings or late evenings in the city, which is the best time (less crowded, better light for photographs, great atmosphere and lighting for after dinner walks).

A van could be useful if you want to explore the countryside, but then you'd probably want to rent it when leaving one city, spend several days exploring the region of your choice, then return it when arriving in the next city. Tuscany is a popular place for this type of exploration, so renting a car after Rome, spending time in Umbria and Tuscany, then returning the car in Florence area is a reasonable plan. Do note that many small towns also do not allow you to drive into them - there are parking areas on the outskirts and then you walk or bus into town. Also, small towns have train connections, too - so you don't need a car for a simple day trip from florence to a tuscan town - more useful for connecting several towns and visiting rural parks/wineries/etc.