All this would be true, in a vacuum. But there was about 100+ years of concerted oppression of certain groups because of their geological origin and skin color. And as for the 'mixed-race' thing - look up the 1924 'Racial Integrity Act of 1924'. That law (and others like it) classified someone as black if they had 'one drop of black blood'. So, plenty of folks would have identified as white, or non-black if they were allowed, but the law literally classified them as black:
According to the Commonwealth of Virginia, Obama would have been black, by law. This law was literally designed (as explicitly stated) to keep the white race pure. This law was in effect when Obama was born, by the way, so if he had visited Virginia as a kid, he would have been legally classified as black, irrespective of skin tone, or percentage of African ancestry (similarly to the way South Africa used to).
So, when we talk about people being mixed race, we are going up against a long history of people using race as a weapon. Plenty of people alive today remember when these laws were still enforced, so just suddenly pretending that race doesn't exist as a powerful cultural construct is to be intentionally naïve.
Histories have powerful effects on people's attitudes and histories. Try wearing an orange sweater on St. Patrick's Day in certain neighborhoods in New York or Boston and tell me how that works out. Why doesn't everybody get along well in Belfast (or Jerusalem, or Delhi, or Kabul, or Dubrovnik)? History, that's why. People have long memories.
When we talk about people being of mixed race we're taking about REALITY, not the rainbows and unicorn farts of fantasyland that deniers pushing "racial purity" BS in our history talked about.
Even Morgan Freeman pointed out Obama wasn't our first black President... was out first mixed race President.
But people tend to simplify such details by nature. I've got German, English, Irish, and Scottish ancestry (that I know about) and I don't go around identifying myself with that level of detail. Most people pick one thing, a few a couple, and that's it for general use.
Most people take great pride in singular identification.
Even without those BS laws, people would have settled into this pattern. It would just have far less stigma associated with it in some circles.