What is typical March weather for central Italy, i.e. Tuscany and Umbria? Would renting a car be an advantage should the weather turn out to be quite wet?
Sure, a car would be an advantage in the rain; you could eat a picnic inside it. A car is pretty much always recommended for exploring Tuscany and Umbria, outside the larger cities of Florence and Siena. When you get to a hilltown or historic area, you will normally have to park outside the Zona Traffico Limitado and walk in from there, unless you have reserved a hotel that arranges permission for you to drive into their location. Hotel and agriturismo options outside of town centers are more likely to have easy, free parking.
Go with the car. If it's raining, you have the option to go to a larger town where there are more things to
see and do.
To visit small towns and countryside a car is always an advantage, regardless of season.
If you stay in Florence a rental car is always a hassle, regardless of season.
So it really depends on where you plan to stay.
If you intend to rent a car for part of your trip (or all your trip) make sure that while you have the car, your accommodations are outside Florence or at least at a hotel outside the city center that provides parking (however why would you need a car while you are in Florence? To keep it parked in the hotel garage?).
Might be a rare occurrence but we ran into problems last year by Volterra. Roads were washed out due to the harsh winter weather and it took a lot longer to get to Volterra from the agriturismo that was recommended to us.... a long way around. Also encountered this same problem at another spot but forget where.
So the plan is to not have a car, but if it starts to rain you will find one to rent?
That's an approach that's new to me.
Since it is March now and your question was in mid February, can we consider your decision made?
FYI, if you are planning this year, I'm in Rome now and froze yesterday in the wind, but your question is about rain not wind, and today I was overheated in my jacket and long sleeve shirt because the sun was out all day - just like yesterday - but the wind was completely gone and the temperature was up from 9C yesterday to 15C today.