The Fall of Rome wasn't a sudden or particularly shocking event. The Empire had been falling apart for some time. Odoacer and Theodoric only officialized what had been going on for centuries.

As I explained in this answer, In the immediate aftermath of the transition from Odoacer's consulship to Theodoric's proclamation as Rex Italiae, a "Roman" was any member of society who spoke Latin as a first language.

There certainly were associations tied to speaking Latin and middle/upper class urban lifestyle, while "Goths" or other "Non-Romans" would have been expected to have arrived in Italy from other parts of the Empire to perform manual labor or serve in the army. Theodoric himself would astutely comment, "A poor Roman plays the Goth, a rich Goth the Roman."

Generally, the cultural pressure and desire to Romanize would lead to there being mostly a number of people speaking Vulgar Latin.