To claim they're all the same is lazy and to anyone observing from outside the U.S - pretty weird. What you described as being Hispanic, the Latin music and all that stuff could equally be applied to Spaniards, and I think Spaniards would be pretty offended if you said they weren't white!
Lazy, or practical? The point isn't to avoid offending people the point is to have a term to describe relevant sub-categories of the population. The term is used both by self-identified members of the population and those who use the classification.
Legitimate European Spanish diaspora in the US is a tiny fraction of the Latin American presence (80k vs 52 million). Odds of Spaniards gathering in sufficient numbers to be mistaken for Latin Americans is very slim. And certainly, the closer you look the more you'll find different patterns in the various Hispanic subgroups (starting with the obvious, like Brazilians not even speaking Spanish).