When Rome conquered CisAlpine Gaul (northern Italy that got occupied by the Gauls), they slaughtered the local population and replaced them with Roman citizens, of Latin origin (they did have a population crisis).
Next, aroudn 50 BC. Julius Caesar invaded Gaul and basicaly committed genocide, with millions supposedly killed (we may never know hte exact numbers but Caesar himself notes down pretty good figures in his Commentaries and whole tribes were wiped out). He replaced these, especially those in the Provence and bordering areas with Latins -- more trustworthy.
By the time of Vespasian, a century later, the Celts had totally given up their culture and had intermingled with the Latins, maybe the old soldiers got htemselves fair-headed brides.
Roman citizenship now was extended to all of Italy and to parts of Provence and Iberia
From this point onwards you are correct, Romans were multi-ethnic. But most had Latin blood.
Oh and yeah, there was no Celtic RACE, they were the same as the Germanics and had the same racial characteristics as the Aryan race to which the Romans too belonged and to the Caucasian race to which a whole lot of people belong.