Columbus' lasting legacy in the US might not have all that much to do with the man himself, aside from the fact that his name came to be associated with the female personification of the Americas in European art—Columbia.
The Renaissance was in full-swing when Europeans first "discovered" then established themselves in the Americas. Classical forms had come back into fashion as artists turned to the Greeks and Romans for inspiration. We see at that time the development of a recurring image of the four continents (Europe, Asia, Africa and America) as four women. In earlier European versions, Europe appears like a Roman goddess, while America is more fruits-and-feathers "Indian princess". Later, the personification of America would also come to look more like a Roman goddess, especially in works created in the New World.
The name
Columbia to describe the New World is first used in Britain during the 18th Century as a way to strategically avoid breaking the law. At the time, it was illegal to put parliamentary proceedings into print, so the press was forced to use thinly veiled pseudonyms for places and people. Samuel Johnson, writing in
The Gentlemen's Magazine at the time, is first credited for using
Columbia to refer to Britain's New World colonies. The name later became widely used in poetry; unlike the word
America,
Columbia could be passed off as a three-syllable word when convenient to the poet. Given the prevalent use of personification as a literary device in the poetry of the time (esp. in regard to countries, continents, &c.), the classically inspired image of America-as-Roman-Goddess and the name Columbia coalesced into the figure of Columbia we know today (e.g.,
here).
Following the Revolution in the Thirteen Colonies, the Founding Fathers and their contemporaries looked to ancient Rome for inspiration, not only in their republican form of government but also in its symbols: the eagle, the fasces, Neoclassical Architecture, etc. The image of Columbia as goddess fit in perfectly (just as the similar image of Marianne caught on with their fellow revolutionaries in France shortly thereafter).
At the time, there was also an outbreak of cancel-the-king culture, and several places and institutions changed their names. For example, King's College in New York became Columbia College (today: Columbia University). Somewhat similarly, Georgetown, MD was subsumed into the new District of Columbia. Several other places throughout the new country were either renamed or newly founded as Columbia (e.g., Columbia, SC). The ship
Columbia (later
Columbia Rediviva), was first launched around this time; it would eventually give its name to the Columbia River and, by extension, British Columbia.
I don't know why Columbus himself became a popular figure in the US's history of itself, but I would imagine the ubiquity of his name facilitated it. I suspect his role may have been overinflated in order to minimize the role of the British monarchy in settling the country as was done with the Mayflower (albeit without the City-on-a-Hill ideological baggage). I would also suspect that he was used by Catholics in general to claim space in American history and not just by Italian-Americans in specific. The founder of the Knights of Columbus, Michael J. McGivney, was the son of Irish immigrants and educated in Quebec, for example.