Purchase Unlocked phone without sim card & purchase sim card in Italy?

Yes, it will.

One question, in picking a mobile provider, is whether you're more interested in calls/texts or in data. Data usage is my priority when I travel. The TIM for Visitors plan worked well for me in my trip to Italy two summers ago.

Are you sure that your phone is only CDMA? Today some providers that make CDMA phones also build in GSM capability (for example, the last several generations of iPhones do that).

If you search this forum, you'll find several threads related to Italian SIM cards. In fact, here's a currently active one:
https://community.ricksteves.com/travel-forum/italy/tim-sim-card-at-rome-airport

I did what you're considering and it worked fine for me. My needs are simple as I use the phone only for calls and have a small tablet for wifi internet. The 30-day TIM Tourist card seems like a good deal, and I found it reliable and easy to use. While the website encourages you to pre-purchase, you might as well just buy one at a TIM store in Italy, there are lots of them.

Yes, buy at a store in Italy...and not at FCO. Buy, if you can, in the city you are first staying. I suggest this becaause I had trouble with the TIM card I bought at FCO and was told by the TIM store in Siena that I would have to go the store-of-purchase to get a refund or exchange. They could not resolve my issue which was charging me much more than the advertised rate for calls to the US.

Live and learn.

Thanks to all for your advice!