This is everything in Italy, not just banking. I remember studying abroad there, it took us a week and a half of “domani” to get the wifi password for the dorm!
1) You could've asked the password to literally anybody in the dorm.
2) Unless this was geological ages ago, by law, all you need to close a bank account is a certified email (or PEC) or a certified letter (you go to the post office with a document and a simple form that specifies where you want your stuff transferred) you don't need to go in person anywhere at all.
The fact that the bank personel in the first bank did not share what you just shared only adds to the misfortune.