You got it perfectly right . Famous case in later years was film producer Carlo Ponti https://en.wikipedia.org/wiki/Carlo_Ponti and actress Sophia Loren (big scandal at the time). Ponti was even banished from entering Italy or he wold be accused of bigamy and put into prison. In the cases of less famous people, sometimes phony marriage acts were produced to avoid social rejection of the new couple and their children.
I'm sorry I was absent during the weekend, 4th round of COVID-vaccine got me down with a fever.
But I have a new fact for you: I found the birth record of a daughter of Giovanni, with another lady singer named Clotilde Albessano:
https://www.antenati.san.beniculturali.it/ark:/12657/an_ua19800595/0MdlObW
Since it was an out-of-wedlock child, she was registered with her mother's last name: Luigia Teresa Giovanna Maria Albessano, born 17th november 1904 in Rome. She was recognized by her father and given his last name in 1907:
https://www.antenati.san.beniculturali.it/ark:/12657/an_ua19803058/5g6MN4p
I'm not sure if this is the same Giovanna that appears in later documents, like his USA immigration and naturalization forms (they're on Familysearch).
Also found Maria's death record in: https://a860-historicalvitalrecords.nyc.gov/
I'll kee searching for the italian and spanish ancestors/sibling, though it's not easy because they're not indexed and both came from relatively large cities with several boroughs within.