Not covering Italy is true for American Express but usually not Visa.
With credit card coverage nothing is ever true across brands like Visa ; the issuing bank and exact card will result in different policies.
So have to always be careful and check your exact terms and fine print.
To the OP's general question they usually have a clause that keeps you covered if it is totally impossible for you to deny any coverage.
Couple of critical points:
1.) Assuming your Visa card does cover you, expect the rental car company to precharge and hold a large deposit, much larger than if you buy their insurances. They are not accepting Visa as the insurer but your card as guarantee of them being paid by you for any damage. After all you are guaranteeing you will cover the damage on the card you present to them.
If you don't have a high limit cushion available it is probably not wise to rent this way.
2.) Should anything happen you need to be the go between and the documentor, keeping all paperwork on repairs, police reports. You have to pay for the damages on your Visa and then once you are home spend a bunch of time submitting a claim and waiting would could be months for Visa to reimburse the charges on your card. I think this is what most don't understand thinking Visa and the rental company in Italy work things out together, no that is on you to get yourself reimbursed.
2b.) Hertz or Avis or any other similar brand in Italy is only loosely related with the brand here in the states, so the US offices will be no help, if you need documentation or most anything post trip, you will have to deal with an Italian office.
2c.) I don't have any personal experience with this but have read some of the smaller players in European rentals like Dollar, Budget have started not permitting Visa only insurance as coverage. This may not be factual and I am certain the big 3 of Hertz, Avis and Europcar will.
3.) Given 1 and 2 are hassles for anyone that values their time or spending capacity it is usually worth comparing how much the insurance from the car companies does cost. if it is not a significant amount than it may not make sense to take on any of this headache yourself and just pay a little more at the counter. if it is a high amount though, might as well use your card's benefits. The Visa cards with the best benefits usually have annual fees, so you are paying for these perks.
I have rented many times in Europe with a Visa card that covers me Have a Chase Sapphire Reserve and a Chase Sapphire Preferred ; sometimes I deny coverage and use the Visa for coverage ; other times I pay extra with the rental for zero liability.
Many of the rental agencies don't quote insurance cost online so I often go to Auto Europe because they do and have cheaper rates for insurance typically.
Last comment from me:
I don't always trust what I hear from calling a large company like Visa or Chase. Fact is the 800 numbers are a roll of the dice, sometimes you get someone great and very knowledgeable other times you get the opposite. It is common advice to call your card issuer for these type of questions but I am not sure that will result in more accuracy on average than just double reading your specific terms and asking questions on forums like this if the terms are not clear.