Kinda, Constantinople didnt much care who was ruling Rome at the time. The east wanted someone who they could control. The east used the goths hoping to give them power they would stop the contant wars. They let "Barbarians" in the gates. The Goths were civilized at the time, Rome had a small recovery with Theodoric the Great. When he died it went all down hill from there. The east took over then you get the Byzantine empire