Were there any tactical advantage to firing arrows in a barrage in medieval times, rather than just firing wildly at will?

Depends entirely on the context in which the archers are being used. Let's look at some probable scenarios :

Pitched Battle : In such a scenario where two armies are facing each other, the archers were usually placed behind infantry, however, you had scenarios where archers would be placed in front of infantry as a screening force against the enemy or they could be placed behind barriers either natural or artificial. Obviously, unless there's a direct line of sight for the archer with the enemy the idea of loosing arrows at will makes little sense. So if the archers are employed as a screening force, they would be free to loose at will, but usually the commanders would prefer to loose in volleys. The effect was twofold. First, suppose you had 3000 archers. Each archer had roughly 10 arrows, meaning you had 30,000 arrows in total. Now, in case of "fire" at will, the archers would target the enemies directly in front of them, taking down the front ranks or at least hitting them, since if the enemy had shields or even padded clothing not every hit would be a kill. Soon your archers would empty their quivers having achieved not much at all, since the enemy would still retain sufficient numbers or indeed almost the same numbers as he had when the battle began.

Now what would happen if you loosed the arrows in volley? A) It would have a psychological effect on the enemy. 3,000 arrows landing on a formation all at once tend to sap at the morale and disheartened the enemy. It slows down the pace of advance and even might render some fresher recruits immobile.

B) When "firing" in volley, the arrows would be fired at an angle. This meant that even if the enemy had shields for all the ranks, the ranks that were further back, would not be able to always defend themselves against the volley and even if 10% of the arrows met their mark, the enemy formation would be shallower, the morale of the nearby troops who would witness their fellow soldiers dead would shatter or be shaken at the least and you would either stop an enemy formation in its tracks or by the third or fourth volley, you'd force this formation on any part of the enemy line, to rout.

Siege : During a siege, the enemy could either be the defenders in the castle or the attackers from beyond. The defenders would in this situation opt for a two pronged approach if their commander knew what he was doing. First, there would be ample archers on the walls, taking cover behind defensive works and loosing arrows with precision and aim at either pre-ordained targets such as officers or royalty or at enemies who had come close to the walls. Meanwhile, there would be a mass of arches in the castle grounds, that would loose volleys in the direction that they were commanded to, for achieving the same effects as mentioned before.

The attackers on the other hand, would inch closer and closer to the castle, (if they indeed are in a hurry to take it) all the while throwing up earth works and artificial barriers to block the enemy arrows. Meanwhile, their own archers would loose arrows from behind these defenses. In this situation, the usage of archers, in mass, is entirely dependent on the context of said siege, but as always if archers are employed in mass, their purpose is to loose in volleys and to achieve the effects as I have listed above.

Sources :

"The Medieval Way of War: Studies in Medieval Military History" by Gregory I. Halfond

"Medieval Warfare: A History" by Maurice Keen, Martin L. Keen

"Infantry warfare in the early fourteenth century" by Kelly DeVries