The Iberian peninsula was occupied by the Visigoths at the time of the Muslim conquest, and they were a Germanic people. Would the 'Spanish' at that time therefore have been Northern European in appearance? Do the Spanish look as they do today because of the Berber conquest?
I can't offer a complete answer to your question, but I can refute your hypothesis: the Visigoths did not drastically affect the Spaniards genetically, linguistically, or religiously. There were relatively few Visigoths, and they did not integrate well with the native Roman Iberians. Just as Iberians continued speaking vulgar Latin (which evolved into medieval Romance and eventually Spanish, Portuguese, Galician, and Catalan), so they continued to be genetically similar to Roman Iberia. Similarly, while the Visigoths were Arian, the Iberians were Catholic, and eventually the Visigoths converted to Catholicism rather than the other way around.