I don't know why the Citibank would be operating like that. Is some special program or protection for that card? Is it a regular credit card? The only time I've had any issues is sometimes ordering things - like tickets - from automated sites in Europe while still in the states. There is a "Verified by VISA" system that I got one card registered in and that's the one I always use for tickets ahead of time. Are you sure they weren't talking about something like this rather than just being in Italy and using your card?
For the last couple of years banks don't even want you to tell them when you are going overseas anymore - they trust their anti-fraud computers to understand when you are in a foreign country automatically. I have 3 cards from different companies without foreign transaction fees and I have never had a problem simply using them anywhere in Italy.
Visa and Mastercard are universally accepted and I would expect to potentially encounter issues with any other card - Discover, AMEX etc.
Tap is by far the easiest way to pay for things because we don't have chip and PIN but chip and signature. Many people report using only their phone to charge things but I would never count 100% on that.
I think there has been some miscommunication or something else is going on with that card because I've never heard of this kind of restriction with any card. If this really is the case that would be reason enough for me to look for a new vendor.
Good luck,
=Tod