The game [archery golf] came to American as a roving game. Both golf and the roving game "apparently" originated independently. Games with clubs and balls were common in Europe way back as were bow games.
Golf [the name originates from words in European languages for club] is a game played with a club. Although well established in our vocabulary, "archery golf" really doesn't make sense because you can't play golf with a bow, by definition it is played with a club.
The original roving game as played in the late 1800s in America, before being taken over by that game played with a club, consisted of roving established paths ending in a target. The course was set so each target took 3 or more shots to reach.
The "hole," for no better word, ended when each player hit the target. Scoring was by number of shots to hit the target and the target score was recorded separately. In the event of a tie in total shots, the total score on the targets determined the tie breaking at the completion of the round. And, there was actually a set of simple rules printed on the back of the scorecard for the earliest course of this type I've been able to validate in St. Louis in the late 1800s.