Vickie: to clarify since I do not disagree with what you stated but it is all how I worded my statement.
With any Visa or any other US credit card (has to have Primary rental not Secondary and only few of the main issued credit cards actually include Italy) ; but for sake of discussion let's say you have one of those cards.
You give them your Visa and don't pay to reduce the deductible, just getting what is provided in Italy at no extra charge.
They charge to your Visa the rental rate and a deductible (anywhere from 600 - 1200 euro either as charge or a hold).
When you return the car ; if no damage they refund/take off hold to your Visa back that deductible so you don't really have any affect unless that amount affects your credit limit while traveling and needing to put other deposits on it like hotel security deposits and the like,
if when you return the car and there is damage, they will deduct the amount of damage from your paid deductible so you pay for it with your Visa. The max they can charge you is that deductible amount. You then have to document everything yourself, paperwork from the rental company and then when you get home file a claim with your Visa card and then your card holder if your claim is approved will reverse any charges on your account so you don't actually have to pay for anything yourself.
For some this is perfectly fine and saves some money.
The point is that requires extra work, it is not handled between the bank and the car rental company, you are in the middle.
The agent even mentions this in your own comments, we have customers that have successfully submitted claims to Visa.
Would be the same if you bought travel insurance that had auto coverage for Italy or if your home insurance covered an Italian rental, etc... You pay and then get reimbursed.
Going with a policy through AutoEurope, Gemut or from Hertz, Avis, Europcar avoids that aspect, if there is damage and you paid for zero deductible you turn it in and walk away (provided you didn't do something outside of the contract as you mentioned that caused the issue). In addition that deductible amount is never on hold with your credit card.
Many renters I think would prefer to pay for this level of extra convenience assuming the cost is not too high.
That is the difference, hope it makes sense. Walk away vs. you pay and get reimbursed later.