America has plenty of mythology. The contemporary mythologies you mention all have roots in the 18th/19th centuries. Typical white mythologies are going to be patriotism (Republican Motherhood for women, which is actually pretty interesting - this idea pops up in children's books and stories for women) and individualism and liberty; that's where George and the cherry tree tale and Honest Abe come in, but also Horatio Alger (which are pretty abdunant 19th century) and John Henry stories. You could also consider many sentimental and domestic works in the 19th century to function as mythologies as they deal primarily with how women are to be guided by God (Wide, Wide, World; The Lamplighter; Ruth Hall; Uncle Tom's Cabin). Slavery gave way to some of the best myths, which appear in songs and other works of art (quilts, for one), but Charles Chesnutt (turn of the 20th century) is a good example of "conjuration" mythologies (see "The Goophered Grapevine" and "The Conjure Woman ") and how those myths are sustained after slavery.
You could also look at how America seeks to build its own mythology by using other forms (and, ultimately, fails in a way), with books by James Fenimore Cooper and his attempt at capturing the expansive and natural America (Deerslayer is probably best in the series though Last of the Mohicans is probably most well known) and Charles Brockden Brown's "Weiland," which was an attempt to take the gothic and make it American (with no castles or landed gentry, it was pretty hard; he tried nonetheless). And the land of purity/deviance brought to mind "Charlotte Temple" by Susana Rowson (1791) which has become this sort of epic mythology complete with gravesites in New England claiming to be poor (fictional) Charlotte (originally published in England, but it is American in theme).
So while Americanists (like me) get chided for our area of study because America is still in its infancy, there is still a lot to work with in terms of mythologies that have nothing to do with Native Americans (although something like "Hope Leslie" by Catherine Maria Sedwick is good because it's revisionist history in the good way by altering the falsehoods about Native Americans). Just like everything else that has to do with America, it has to be tweaked and reconfigured. But perhaps the most inherent Americanism in terms of mythology is that Americans have to make it new and make it particular to them.
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You could also look at how America seeks to build its own mythology by using other forms (and, ultimately, fails in a way), with books by James Fenimore Cooper and his attempt at capturing the expansive and natural America (Deerslayer is probably best in the series though Last of the Mohicans is probably most well known) and Charles Brockden Brown's "Weiland," which was an attempt to take the gothic and make it American (with no castles or landed gentry, it was pretty hard; he tried nonetheless). And the land of purity/deviance brought to mind "Charlotte Temple" by Susana Rowson (1791) which has become this sort of epic mythology complete with gravesites in New England claiming to be poor (fictional) Charlotte (originally published in England, but it is American in theme).
So while Americanists (like me) get chided for our area of study because America is still in its infancy, there is still a lot to work with in terms of mythologies that have nothing to do with Native Americans (although something like "Hope Leslie" by Catherine Maria Sedwick is good because it's revisionist history in the good way by altering the falsehoods about Native Americans). Just like everything else that has to do with America, it has to be tweaked and reconfigured. But perhaps the most inherent Americanism in terms of mythology is that Americans have to make it new and make it particular to them.