It should be the opposite, since the Goths were Christianized before the Romans were at large… in particular those that participated in the Great Migration. Those Germanic Christians are also why so many Christian holidays are molded after German holidays, often moreso than Latin ones (the only Christian holiday with a direct Latin equivalent is Christmas with Saturnalia, but even Germans had celebrations around that time, Easter is also very much modeled after the Germanic spring celebrations).

Christianized Germanic were the reason the Roman Empire became Christian after all, since they made up the backbone of the army under Constantine I, so become Christian was a smart way to ensure their loyalty.

While Christianity had a decent following in Rome, it didn't explode in popularity until (Christian) Germanic tribes settled in the Empire.

Just a little known factoid about the rise of Christianity in Europe.

You're right that the Goths were largely Christianized at that time, but you're missing an important detail – most of the Goths (along with the Vandals) would have belonged to the Arian church rather than the Catholic/Orthodox church the Roman Empire sponsored, which would later dominate Europe post-Charlemagne.

Also, I'm not entirely sure about Rome not being largely Christian by the time it fell – I knew many of the rural areas were Pagan, but the imperial church had a lot of power in the cities at the time (mostly in the East, but to a large extent in the West too). When the West fell, it was largely loyalty to the Catholic/Orthodox church that distinguished between Romans and Germanics. The Vandals were quite repressive to their Catholic/Orthodox Roman subjects . Their Gothic counterparts were more tolerant, but there was still tension between the Arian Germanic elite and the Catholic Roman population, as seen in the revolt in Liuvigild's Spain . So it's likely that many Romans were already Christian

I'm also not sure about Constantine converting to appease his Germanic Christian soldiers, seeing as Ulfilas only began evangelizing to the Goths during the time of Constantine.