Did Rome Fall Because of Immigration?
I doubt that the favorite historical analogy of the anti-immigration Right holds water.
Rome's economy can be tracked by measuring the lead content of peat bogs in Spain. Those bogs show that Rome's GDP (and probably other measures of its power) was well down from its peak even before Valens invited in the Visigoths. The barbarian invasions were a consequence of Rome's weakness, not a cause.
It's unlikely there were very many Gothic immigrants anyway, or at least not enough for much of a linguistic impact. The languages spoken in most of Western Europe are derived from Latin, not German.