You might be interested in this answer posted by u/hergrim recently, where he explains some of the issues with the battle and how the charge made by the Dothraki was more or less completely at odds with typical nomad cavalry tactics.

That said, assuming a gap that large did open up in the lannister line "for some reason" and the Dothraki had managed to train their horses to charge straight into fire which just moments earlier we saw had been hot enough to burn a man to a crisp, then it would have almost certainly meant the end of the Lannister army. It was almost impossible to expect a line of infantry not to run or stay focused on what's in front of them if they know that there are enemy horsemen running around, unchecked behind their backs. This is especially problematic since the the Lannister line is so thin, and only one rank seems to be carrying spears. Early modern military authors tended to be more comfortable with at least 5 or 6 ranks of pikemen to defend against cavalry, and far, far deeper if they needed to properly defend from all sides. With only one line of spears it would only take a couple of soldiers becoming injured or losing their nerve before gaps started to appear.

If anything it felt like the battle showed the Lannisters performing a bit too well, since the vast majority of the dothraki army apperently vanished after the initial charge so that they could show a bunch of scattered, one-on-one duels for the next several minutes.

You are correct in the argument if you both are assuming the hole in the line remains. The shield wall tactic that the Lannisters were using in the battle relies on the rigidity and continuity of the line. When gaps, even small ones, open up in the line, enemy cavalry, especially experienced veterans, could easily exploit them to essentially break the formation and destroy the entire line. Once the horsemen are through behind the line, the strength and coherence of the shield wall dissolves. Now the warriors forming the wall are under attack from both the front and the back and, barring some really spectacular strength of will, panic is not far behind. Of course, panicking infantry are easy targets for cavalry of any era. We can cite a number of examples to back up this general idea.

The first example is the Battle of Hastings. The Saxon shieldwall at Hastings proved impervious to all Norman cavalry and infantry assaults during the initial stages of the battle. However, when the Norman forces started to fall back in disarray, Saxon troops broke from the shieldwall to pursue them. Realizing that this was weakening the enemy line, William had his troops stage feigned retreats a number of times to draw enemy forces away from the wall and into open ground where they could be slaughtered. This ultimately weakened the wall to the point where gaps began to appear that could not be readily filled by more than a handful of men, which William's cavalry were then able to exploit. Once they were through the gaps, the shield wall dissolved and the Saxon forces routed (except for, according to some sources, the personal bodyguard of Harold, which purportedly held formation around his standard and died to a man).

Another example we can draw from is the Napoleonic Wars. Though not a shield wall, the cavalry square of the Napoleonic Wars was a similar formation to what the Lannisters used in that it involved tightly packed ranks of men using bayonets as spears to fend of cavalry attacks with the added benefit that some ranks could shoot instead of stabbing. The cavalry square was considered impregnable by cavalry IF it held formation and no gaps appeared. At the Battle of Garcia Hernandez during the Peninsular War, a brigade of dragoons made the decision to charge a French infantry square. The French fired their volley late, but would most likely have driven off the charge, except for the fact that one mortally wounded horse tripped, fell, and slid like a dead weight into the side of the square. Though it only disrupted less than a dozen men, the cavalry were able to exploit that small gap in the formation, pushing into the center of the square which quickly broke, resulting in the death or capture of almost all the infantry involved.

In short, the gap makes all the difference. While it's possible that, given their superior numbers, the Dothraki would've broken the wall by sheer weight of men and horses (this is probably why Bronn tells Jaime to book it as soon as the Dothraki horde comes into view), having that gap made it infinitely easier for them to do so.