Perception is the reality for everyone. What in the world makes you think that just because your kids have had a different experience than Obama that means that Obama's reality is only based on his perception while your kids' reality is based on something more objective??
Speaking from experience, it beats the alternative.
That's because there is very little there other than the white privilege. There's a bit of cultural content, such as the use off the English language and certain popular holidays, but even those are just expressions of white privilege. "White" is a status that certain groups obtain in this culture, which affords them privilege over others. Which also explains why white is seen negatively, because it is by its very nature based on inequity and oppression. Groups such as the Irish and the Italians and Poles were seen as "not white" for generations and only attained "whiteness" post-WWI. Members of other groups such as Asian Indians have sued in order to be recognized as "white" and lost.
If you want to explore culture, explore your ethnicity and nationality, not your race. Race, all "races" are the result this white privilege and the other groups orienting around that concept. There was no concept of "Asian" identity until colonialization. The same with African and Latin-American and Native American.
Let me put it this way:
Chinese pride - great.
Mexican pride - fine.
Irish pride - dandy.
Indian pride - super.
German pride - swell.
Jamaican pride - wonderful.
Norwegian pride - cool.
White pride - NO.
If you want to explore your European ancestry and culture, that's fantastic. That takes you out of the center of white privilege and into the periphery of being an ethnicity, like everyone else, with "American" at the center. True, multi-cultural American, not white American as is the case now. (And yeah, multiple ethnicities are fine.) That's why Zephyr and I get along. Pride in one's European ancestry is not the same as "white pride." It's a shame that many white Americans don't see that and try to do the "I don't see color" thing. Or the "We're all the same" thing.
As for African-Americans, their ancestors were forcibly cut off from their cultures and transplanted over here. Many descendants do not know their exact ethnic heritage and often times are mixed anyway, due to masters raping their slaves. (Which makes the OP quite ironic, as the vast majority of those whom we call "black" in this country are already of mixed race.) Many Africans do not see African-Americans as kindred. For many blacks, the only culture that they have is African American or "black" culture. Tho some are exploring West African culture, which is where most of the slaves came from.