The replacement with of "men behind the throne" with Visigoths might be significant. The Visigoths controlled much of Spain and Aquitaine in France, so if their control of the Emperor is also part of this equation, then you've reunited Spain, Italy, and Southern France under the Imperial umbrella. With the strength that the Visigoths bring to the table, they could have a pretty secure grip on the Western Empire (such as it is), and prop up their Imperial puppets for some time.
By using the authority of the Western Empire, they could create themselves as the central power in the West. If the Visigoths follow this path, then they are investing themselves in the position of the Emperor, and thus the "Western Empire" won't end with the regalia being sent east by Odoacer.
On another note:
If the Visigoths are the power behind the throne in the West, then something needs to be done with Ostrogoths. iOTL Theodoric invaded Italy in 488, at the behest of the Eastern Roman Emperor, in order to overthrow Odoacer's rule. With the Visigoths in control, this event, and the events that led to it, will be butterflied.
A different, and very interesting option, for the Ostrogoths is North Africa. Geiseric died in 477, and with his death the Vandals immediately went into decline. The Vandals are not friendly with anyone really, the East Romans hate them, the Visigoths don't like them much, and so having Theodoric lead his people into North Africa could be a very cool option.
The plan, as I see it, would involve a joint Visigoth-East Roman expedition. The East Roman would only need to pledge ships, as the troops will be provided by the Ostrogoths. The Vandals were able to fight off an earlier invasion, but that was under Geiseric. This invasion will be successful, and put the Ostrogoths in control of North Africa.