Just because the ethnic distinction disappeared doesn't mean the Visigoths weren't integrated to some degree or another into post-Reconquista identity. Sure, the political continuity wasn't recognized because the rulers were Germanic and the Gothic language became antiquated and died. But ethnically speaking, the
people of Spain by the time of the Reconquista were a fairly homogenous (for the time period) mix of Gothic and Latin heritage. Not to mention, Spain's whole Catholic agenda, not just under the HRE but unification under the Catholic monarchs, was set in motion by the Goths, what with the conversion of the monarchy to Catholicism and the Councils of Toledo defining the last century of Gothic rule.
Sure, the modern Spanish identity doesn't like to think about its messy origins as a Germanic-Roman offshoot taken over by West Arabia. But it's still there, and arguably the only thing about the Goths that was dissonant with and never integrated into the Kingdom of Spain was their language, which was quickly relegated to liturgical use much like Greek or Latin. There are large portions of the Byzantine Empire that don't identify as successors to the Byzantines, either, despite owing a large part of their upstart to Byzantine influence (see: Bulgaria, Turkey). But Bulgaria I don't believe was ever ethnically Byzantine like the Spanish were Gothic?
I'm not arguing for a second Gothic Spanish leader, here, because I think regardless of the basis the idea of a Germanic-speaking Spain leader is just too dissonant and esoteric. Not to mention that by rights any Gothic representation should probably be as the unified Gothic Empire under Theoderic. I'm just noting that like the Bulgars, who were originally Turkic but took on a more Slavic identity over time, identity is fluid. Bulgaria still cites to and is proud of its Turkic heritage, and there is nothing shameful about associating with the Visigothic Kingdom. There really isn't much reason other than centuries-old propaganda supporting the idea that Hispania did not exist before Spain. Because it totally did. It was called Toledo.
Though now that we have an Occitan-speaking leader for England, I guess weirder things are now possible. Theoderic leads Spain and Italy!