I recall having the same issue you describe the first time I tried to make a purchase. As I recall, I was able to eliminate this issue by selecting "Go on without log in" in the AUTHENTICATION dialog. (I just tested this again now and it worked.)
I just returned from 3 weeks in Italy, and used the Trenitalia site (as well as the mobile app) often to purchase tickets. A few I bought while in US before flying to Italy; the others I bought while in Italy. Some were 2nd class assigned seating, some were unassigned seating, and one was 1st class assigned seating. I DID NOT REGISTER (so thus when paying selected "Go on without log in" under the AUTHENTICATION dialog at top of page, though one still needs to enter name and email in this AUTHENTICATION dialog; an error message will alert you to the need to enter this information). Then I paid with credit card.
If you are traveling with two or more people, a discount you might be able to get (as we did) on some stretches is the "insieme" (joint) discount. If available, it will come up as an option.
My wife and I were pleasantly surprised by Trenitalia. We used their service when travelling from Sicily to Naples (which included riding a train ferry between Sicily and the mainland), Rome to Orvieto, Siena to CInque Terre, and Cinque Terre to Milan. On one stretch (Siena to La Spezia) we had two train changes, with 5 minutes and 9 minutes to make the two changes. Even with our luggage we had no problem catching the connecting train. The conductors we encountered were unfailingly polite. In most cases we just showed our pdf ticket on our cellphone; in one case we had a printed copy. Only one of the many trains we rode experienced a delay (30 minutes); I couldn't understand the announced Italian explanation as to the cause.
It was wonderful to buy the tickets in advance and avoid the lines as well as the need to validate the tickets.
BTW, we also bought advance tickets online for the italo high-speed train from Naples to Rome, and for the Malpensa Express from Milan to Malpensa airport. No issues.
Have a good trip.