The idea of longbows being used in the Revolution is intriguing, if only because of the superior rate of fire archers could bring to the black powder battlefield. The comments about draw weight of the English longbow are accurate, but those had to propel bodkin-tipped arrows that could penetrate armor. Personal armor wasn't unknown in the British army by the 1770s, but it was limited to officers for the most part. (Some heavy cavalry units still used armor IIRC.) Common soldiers were unprotected.
So rather than 150-pound longbows that required years of training to shoot accurately, archers in the Revolution could theoretically get by with lighter bows -- say 75-pound bows that were well within the strength range of any New England farm boy. And massed units of archers would not have to worry so much about individual accuracy. They could be trained to fire at certain angles (for certain ranges) at command in the same way (and likely less time) that a soldier could learn to load, aim, and fire a musket. So archery becomes less a matter of individual skill and more a matter of learning to keep the pointy end of the arrow toward the enemy.
Which still leaves the question of why the concept of archery warfare would have survived into the latter 1700s. Or perhaps have been rejuvenated by circumstances.
1. The Continental Congress recruits warriors from friendly Indian tribes and forms them into archery units.
2. A shortage of firearm manufacturing facilities above the village smithy level and lack of reinforcement and supplies from France.
3. A persistent and intractable shortage of gunpowder and/or the desire to reserve gunpowder for artillery.
4. A desperate commander, perhaps cut off for months in northern New York or somewhere in the western territories without access to resupply, falls back on memories of a childhood among local Indians and stories of Robin Hood to turn part of his troops into archery units as stores of gunpowder and spare musket parts begin to run low.