The solution to this problem proposed by Adrian Goldsworthy, and to the question of how a convex could've prevented a line from breaking (and personally, if it works why didn't everyone else use it) can be seen in the deployment here. As you can see, if the Libyans were deployed both between and behind the cavalry and Spanish/Gaellic infantry, then they would be blocked from view of the Romans simply by the bodies of men and horse in front.

In his previous battles with the Romans, Hannibal saw that even when put into a disadvantageous position, the main thrust of the Roman infantry were able to break through the Carthaginian line. He therefore advanced his center first into a convex to deal with this problem. But whereas scholars before Goldsworthy assumed Hannibal's convex also prevented the Carthaginian line from breaking, Goldsworthy's hypothesis is that Hannibal turned this problem to his advantage. The Romans would naturally crowd towards the center where the fighting was, and troops would be committed to where a breakthrough seemed imminent, also the center, as it retreated. The position of the Libyans also served to stablize and reinforce the Carthaginian infantry on the wings to make sure they do not break. Instead of trying to prevent the Romans from breaking through the center, Hannibal counted on the fact that they would. The Carthaginian center breaking is reported by Livy in the infantry battle:

At first they were evenly matched in strength and courage, as long as the Gauls and Spaniards maintained their ranks; but at last the Romans, by prolonged and frequent efforts, pushing forward with an even front and a dense line, drove in the wedge-like formation which projected from the enemy's line, for it was too thin to be strong; and then, as the Gauls and Spaniards gave way and fell back in confusion, pressed forward and without once stopping forced their way through the crowd of fleeing, panic-stricken foes, till they reached first the centre and ultimately —for they met with no resistance —the African supports. These had been used to form the two wings, which had been drawn back, while the centre, where the Gauls and Spaniards had been stationed, projected somewhat. When this wedge was first driven back so far as to straighten the front, and then, continuing to yield, even left a hollow in the centre, the Africans had already begun a flanking movement on either side, and as the Romans rushed incautiously in between, they enveloped them, and presently, extending their wings, crescent-wise, even closed in on their rear. From this moment the Romans, who had gained one battle to no purpose, gave over the pursuit and slaughter of the Gauls and Spaniards and began a new fight with the Africans. In this they were at a twofold disadvantage: they were shut in, while their enemies ranged on every side of them; they were tired, and faced troops that were fresh and strong.

You can see Goldsworth's interpretation of what happened here. Once the Romans broke though, as they've always done, in their pursuit they would expose themselves to the Libyans on the flanks which were originally blocked from view. The Libyans would then naturally turn inwards and sandwich the Romans, who would be crowded and disorderly from pursuing the Carthagnian center (and as Livy reported, tired). This would also give Hannibal and Mago the opening to rally the Carthaginian center and re-commit them to the fight.

I like this interpretation for many reasons:

1. It's logical for us. The Romans broke through the Carthaginain line before so no reason to reason to assume they couldn't again.
2. It's logical for the commanders. Both sides worked off what they experienced in previous pitched battles, the Romans to focus on breaking through the center which is something they knew they could do and Hannibal to use that against them. And Hannibal, knowing he was best at ambushes, where the terrain gave him no place to hide an ambush instead hid one with his army.
3. It explains how the convex worked more logically, as formation of men don't function like architectural arches, and men who fell back from the fighting should break.
4. For the purpose of this question, it means the Romans weren't idiots who left the Libyans on the wings unengaged. The Libyans were held in reserve and were blocked from view by the front line. They were indeed "hanging around and eating popcorns" unmolested until the Romans broke the Carthaginian center, because they were reserves.
5. It is less stylish and more chaotic. Battle is chaotic, and the perfect encirclement traditionally proposed seems a bit too perfect.
6. It is simpler. Again, battle is chaotic, and armies of the time had very little training and no wireless radios to communicate with each other. The fewer orders you issue to achieve the desired results, the more likely it is to succeed (still the case today). Rather than Hannibal and Mago needing to do a prolonged, controlled fall-back while engaged, they can let the battle play out naturally once joined, and unlike some interpretations the Libyans don't need to march forward and wheel in. They would be fairly stationary, "lying" in ambush until it is sprung.
7. It better matches Polybius' casualty count for the Carthaginians. Though Polybius doesn't report the Carthaginian line breaking, he report the Carthaginian casualty as four thousand Celts, fifteen hundred Iberians and Libyans, and about two hundred horse. In other words, the Carthaginian casualties were overwhelmingly of the center. We know most casualties in battle are taken in the pursuit when the line breaks. If the Carthaginian center did not break, then the distribution of casualties should be more even.
8. It gives more room for defeated Romans to escape. Livy reported not an insignificant number of Romans escaping the battlefield, and most historians seem to agree that is more accurate than Polybius' complete destruction. If so, then this interpretation which would naturally have significant gaps at various points in time would better allow this, and the Carthaginian army would not need to try to slaughter a completely encircled Roman army fighting to the death, which would have meant huge casualties for the Carthaginians as well. Rather, in this interpretation most of the Romans would likely try to escape through the various gaps in the encirclement, resulting in a rout that the Carthaginians could cut down.
9. It explains why Hannibal's convex isn't more widely used. It doesn't prevent the line from breaking, but rather encourage it to break at a designated point. A huge break in the line is a huge risk that, even if successful, result in significant casualties. Most commanders (including Hannibal himself) in most circumstances would likely rather fight the battle evenly than take that risk, and if victory is achieved without such a break casualties for the victor would be minimal and far less than one achieved with. But many commanders do know the value of having reserves in place to plug a break should one happen, and defense in depth was used throughout history.