No. And that's not a particularly helpful comment, since you can't prove the converse.
Fair enough.
The advantages of volley fire outweigh any advantage you might gain in rate of fire. Men firing as they wish are going to be inaccurate. With peer pressure and direction from officers you can make everyone hit the same spot. Why trust individuals to judge their shots when you can control their fire to ensure that everyone hits the advancing enemy? There is also this impression that volley fire is somehow significantly slower than firing at will. If anything everyone will be faster by proceeding in a coordinated and instinctive manner, doing as they have been trained rather than being under pressure to make difficult decisions on their own.
Men were drilled because they needed to do as ordered, it was the officers who needed to be able to think independently. Allowing individual decision-making might increase rate of fire for the more skilled, confident, and independently-minded individuals, but for others it would slow their rate of fire to below average or make them so inaccurate as to be useless, or reducing the effectiveness of the fire while increasing its volume. The structure of armies for all time has always revolved around lowering the total amount of decisions that need to be made.
My point is that a higher rate of fire does not necessarily lead to more effective fire.
I can't speak for fire at will commands, as I'm not even sure if they existed when or how they were used
But to look at Agincourt where 6,000 English soldiers defeated 20-30,000 French soldiers with volley fire can show quite accurately when volley fire works just perfectly.
The English army had been marching for quite some time, suffering from starvation, dysentery, homesickness and constantly being outmaneuvred by a superior French force that was growing and whose commanders were competently maintaining their tactical advantages against the English. The French were aware of the danger of longbowmen, and aware that the English had sizeable numbers of them, thus the French commander Boucicault favoured simply starving the English army to surrender instead of risking open battle. If the French had known of the 6,000 English soldiers, 5,000 of them were Longbowmen from Wales and England, they may have attempted a more risky assault instead of their conservative approach which pretty much sealed the deal (though then again, French experiences with archers were that they were relatively worthless, so they may not have listened to Boucicault anyways, as part of the problem with the French army was Lords eager for glory ignoring the wisdom of experienced soldiers). They encamped at Agincourt, blocking the road to Calais, with the supply issue being quite shite, this forced the English to fight on French terms. The site the French chose was altogether quite narrow, with the ground being heavy clay (slowing their cavalry down), the French wearing much heavier armour after it had rained all over that heavy clay (heavy, sticky mud + heavy plate = slow, laborous charge), bounded by hills and forests.
The French delayed battle for 3 days, at which point King Henry in charge of the English army advanced to the narrowest point of the battlefield, setting up stakes before ordering his longbowmen with the heaviest bows to fire galling arrows into the French lines. These arrows were not meant to do much beyond wound, frighten and disorientate French horses and soldiers, which the nobility (making up the cavalry) found highly dishonourable and insulting. The French commanders Charles d'Albert and Boucicault were both experienced soldiers and good commanders, but they were not considered of high enough rank to be worth respecting, thus the cavalry ignored their commands and amassed to charge the dishonourable English longbowmen. Having already been attacked unprepared, the cavalry then attacked unprepared out of anger, with only 500 knights able to charge out of the 2,500 they had at the battlefield. In order to salvage a tremendously deteriorating situation, the infantry were ordered to follow up the knights. The heavy infantry, already slowed by the mud, were now even further slowed by the mud kicked up by the charging cavalry. The longbowmen held their arrows en masse until the cavalry were within 220 to 240 yards, at which point they loosed arrows into the cavalry.
5,000 longbowmen vs 500 knights, horses panicked, riders were thrown from their saddles, were inflicted with terrible wounds, the horses thrown back - trampling right into their own advancing infantry. The French army had crossbowmen and longbowmen of their own, but they had been placed behind the French footmen because they were unwilling to use them (sharing the same line as commoners and servants the Lords didn't really care to use), meaning the battle's outcome now depended on the fatigued heavy infantry. The heavy infantry were injured greatly by the volleys of longbowmen fire, but the real killer was their own march, trampling over and crushing their own fallen infantry in the drowning mud. This crushing march eventually reached the English lines and even managed to push them back a bit, but by then a counterattack was all that was needed to overpower the tired and wounded, capturing thousands (and momentarily executing many prisoners due to reports of French raids on their baggage train). The French were by then so densely packed that not only were they killing their own by falling upon one another, but they could hardly use their own weapons effectively in hand to hand combat. The English flanks continued pouring point blank fire at the packed infantry, which may have even managed to severely injure even the heaviest armoured nobleman, but the most significant thing is that at this point the French infantry were so exhausted they were pretty much done for.
With visors, a crush of human bodies and the long march under thousands of arrows, many suffocated to death in their own armour, or when knocked down no longer had the strength to lift themselves from the muck under the weight of so much armour.
I'm not really convinced then that the issue here is accuracy, speed, or independent thinking. Agincourt displayed both, and neither hampered accuracy or speed, and the longbowmen on the flanks were just as accurate as the core. I would suppose speed and accuracy are a matter of how well trained and disciplined the troops are, after all, your volley is only as good as your leadership and your soldiers' training (Henry ordering the first volley fire as the knights entered the furthest effective longbow range for example). Moreover, a shoddy archer will not be able to shoot where they are commanded to shoot if they are a shoddy archer, you don't really get better accuracy from an inaccurate archer - training is paramount (as the saying went, you trained a Welsh longbowman starting with his grandfather).
I suppose the greatest difference is that since you're not going to cause much in the way of casualties on heavily armoured infantry or cavalry either way with your arrows, with a volley you cause the most wounds and most importantly - panic. I reckon the psychological impact of a rain of thousands of arrows is more demoralizing than a pattering of arrows
Though I suspect you would want a pattering of arrows if you wanted to suppress enemy archers, that last bit is speculation I'm just completely guessing on tho