The Goths stayed only 3 days, I think. I think there is a source which claims the Vandals took the stuff from Jerusalem when they sacked Rome. Of course that may or may not be accurate.
I think it’s pretty obvious most of the loot was melted down and spent. It is, of course, possible that the emperors kept one or two trophies around.
YOu are right that the Gothic sack was mild, as these things go, compared to that by the Vandals.
Both were restrained compared to that carried out by Imperial troops in 1527.