Probably one English longbow archer wouldn't have made much difference in a battle, because with all his training, the best he could do at 200 yards was get it close, and this wasn't horseshoes. 100 of them shooting at the same time was a different story, however. With 100 of them getting close, a few of them are going to hit, especially since the enemy usually marched or rode in close ranks.
Incidentally, at that time it was mandatory that fit Englishmen practice with the bow on Sunday afternoons, but illegal to practice at ranges of less than 100 yards (for fear of encouraging poaching).
My feeling about target panic is that there are many ways to cure it, so you should try whatever occurs to you. If the first thing you try doesn't work, try something else.
I do believe that not trying anything isn't a good plan. For me, that just made it worse over time. In fact, they say that anything you try will work, for a while. A famous archer once suggested folding up a dollar bill and putting it in your shoe under your left heel. He guaranteed that it would cure target panic....for one shot.