This is a good video on the Goths. https://x.com/GothamGirl11/status/1709928682914259303
The Goths were not comparable to today's barbarians because they wanted to become Roman. Their invasion was due to lack of pay from their merc work and the Ostrogothic Kingdom of Italy had little to no major social changes, just as the Vandals contrary to proper belief changed little when they invaded Roman northern Africa
I believe different Goths didn't want to convert to Christianity. Therefore Rome tortured them until the Goths had enough.
But in the end, King Recared converted after his brother, Saint Hermenegild, died for his faith.