Second, who said any attempt at equality has to be made that way? There were historically (and still are) lots of societies both valuing equality and actually way more equal than modern US but with very different structures and/or beliefs.

From the Amish and hippy communes, and ancient Spartans, to modern nordic countries.

To quote Anacharsis from 25 centuries before: "[your laws] are no different from spiders' webs. They'll restrain anyone weak and insignificant who gets caught in them, but they'll be torn to shreds by people with power and wealth."

Or to quote Anatole France from last century: "In its majestic equality, the law forbids rich and poor alike to sleep under bridges, beg in the streets and steal loaves of bread."

Same goes for economic accomplishments of one's family and the opportunities their offspring get in life: http://thewireless.co.nz/articles/the-pencilsword-on-a-plate

You might not want people to be exactly equal economically, but you don't want huge discrepancies either for those reasons.

But of course the right, and especially in the US, has a naive child-like view of a world of equal opportunity were individual work and merit make all the difference. One might as well believe in magic fairies...