This is what Britannica says:
"Football, any of a number of related games, all of which are characterized by two persons or teams attempting to kick, carry, throw, or otherwise propel a ball toward an opponent’s goal. In some of these games, only kicking is allowed; in others, kicking has become less important than other means of propulsion."
"The institutional basis for the most widely played of these new games was England’s Football Association (1863). References to “Association football” were soon abbreviated to “soccer.” Graduates of Rugby School, accustomed to rules that permitted carrying and throwing as well as kicking the ball, played their game, rugby, under the aegis of the Rugby Football Union (1871). When Thomas Wentworth Wills (1835–80) combined Rugby’s rules with those from Harrow and Winchester, Australian rules football was born. In the United States, rugby was quickly transformed into gridiron football. (The name came from the white stripes that crossed the field at 10-yard [9.1-metre] intervals.) Although Gaelic football is similar to these other “codes,” that game was institutionalized under the auspices of the Gaelic Athletic Association (1884) as a distinctively Irish alternative to the imported English games of soccer and rugby."
"Gridiron football, version of the sport of football so named for the vertical yard lines marking the rectangular field. Gridiron football evolved from English rugby and soccer (association football);"
These parts of the article showed how important kicking was in American football:
"(Snapping the ball with the hand became legal in 1890, though snapping with the foot continued as an option until 1913.)"
"Camp was also responsible for having 11 players on a side, for devising a new scoring system in 1883 with two points for a touchdown, four points for the goal after a touchdown, and five points for a field goal (a field goal became worth three points in 1909, a touchdown six points in 1912)"
Also, the original (1863) laws of association football (soccer) had a fair catch rule, a knock-on rule, and the goals were similar to rugby goals:
"A Fair Catch is when the ball is caught, after it has touched the person of an adversary, or has been kicked or knocked on by an adversary, and before it has touched the ground or one of the side catching it; but if the ball is kicked from behind the goal line, a fair catch cannot be made."
"Knocking On is when a player strikes or propels the ball with his hands or arms."
"the goals shall be defined by two upright posts, 8 yards apart, without any tape or bar across them."
"A goal shall be won when the ball passes between the goal posts or over the space between the goal posts (at whatever height), not being thrown, knocked on, or carried."