It'a a (partially) false premise. Cricket was popular with Americans (at least those with high social status) long after the Civil War.
While the increasing popularity of baseball did present a formidable
challenge to American cricket, the two games existed comfortably
side-by-side throughout the 1850s and 60s. It was not uncommon, in
fact, for cricket and baseball teams to challenge one another to
matches in their rival’s sport. (source)
However much the American elite enjoyed playing cricket, the sport never developed the kind of infrastructure that leads to a mass fan base: frequent matches with large crowds and intense rivalries. Baseball, however
. . . was later blessed by a cadre of brilliant entrepreneurs
determined to make it the “nation’s pastime.” One such person was A.
G. Spalding, star player, manager, league organizer, and sports
manufacturer. To call Spalding an impresario or a marketing genius
would be a bit of an understatement. He engaged in every part of the
game, from promoting star players and intercity rivalries to
squelching nascent efforts at labor organization among players. (source)
Even as late as 1911, Spalding saw cricket as the competition. Accordingly, he used its genteel associations as a way to promote baseball:
I have declared that Cricket is a genteel game. It is. Our British
Cricketer, having finished his day’s labor at noon, may don his
negligee shirt, his white trousers, his gorgeous hosiery and his
canvas shoes, and sally forth to the field of sport, with his
sweetheart on one arm and his Cricket bat under the other, knowing
that he may engage in his national pastime without soiling his linen
or neglecting his lady . . .
Cricket is a gentle pastime. Base Ball is War! Cricket is an Athletic
Sociable, played and applauded in a conventional, decorous and English
manner. Base Ball is an Athletic Turmoil, played and applauded in an
unconventional, enthusiastic and American manner.