I question the "much more effective at any range" statement. Archers in the 16th century are a dying breed. Archers fire shots per minute. Arquebusiers fire minutes per shot. However, you are comparing archers versus arquebusiers without the context of changes in warfare. Without bayonets, arquebusiers were tied to accompanying pikes, as are archers. Armor had increased for infantry. The issue of armor penetration had changed the playing field. Shot penetrated, or deeply dented, armor better than bolts or arrows. In any case, the real issue for any missile troops is determining range to target. Even the muskets of the 19th century had rather curved trajectories.
By 1772, armor had nearly vanished from the battlefield, as had pikes. The relative lack of cavalry in the ARW aids the use of even mediocre archers for light skirmishing, hit and run raids, ambushes and similar small scale warfare. I cannot see an establishment of line infantry archers from the military leaders. However, a regular regiment of archer skirmishers would have been an obstacle for any British force forming up near American lines. Plus, no obscuring smoke.