If you're talking about gun shields in a similar context to your image, then it's similar to the reasons why armor in general fell out of favor. Firearms rapidly overtook armor through the centuries. Bulletproof armor needed to be thicker as time went on, and that meant it became heavier. Protection would gradually be limited to only the most vital areas (see Cuirassiers, for example), and even then the protection offered was limited enough that many forces went without armor entirely.

Given that, it's no surprise that gun-mounted armor would have limited utility and would be limited to niche uses. A shield on a gun like in your link adds significant weight and makes the gun unweildy while not really providing significant protection. In the case of the pistol in your link, it's even worse when you look at the wider context of how pistols were often used at the time. As with any military firearm, factors outside of firing are often more important in the design process than the function of the gun itself. Cavalrymen would often carry a number of pre-loaded pistols on the saddle of their horse to be able to fire off several shots in rapid succession. But this gun shield makes the gun impractical to store and carry. So sure, the shield may offer some marginal protection against a bullet, but how is a cavalryman supposed to store it as he rides? After all, soldiers are always moving (on- and off-battlefield) far more often than they're actively fighting.

So with all that being said, why do we see gun shields pop up again at the end of the 19th Century on artillery pieces? After all, plenty of what I said above applies to artillery - gun shields are heavy, and if an eighth-inch of steel isn't stopping a rifle bullet then what protection could it provide from an artillery round? The answer is shrapnel. The nature of artillery fire changed drastically at the end of the 19th Century, with new high explosives and improved fuses making artillery shells particularly effective against unarmored area targets. Unlike before, where a solid shot would be particularly devastating to a small section of the line, high explosive and shrapnel shells could devastate a much wider area. Artillery became particularly vulnerable to counter-battery fire - it lacked mobility, and shrapnel now meant that shells only had to be close, rather than hitting the gun or crews directly.

However, shrapnel lacks the power that a bullet or solid shot has. Much like a shotgun blast, it covers a wider area, but the individual pieces have less energy than a typical bullet, and thus are more feasible to stop. All of a sudden, gun shields become relevant again. A thin piece of metal may not be enough to stop a rifle bullet, but it'd provide protection from near-misses from artillery shells, which lets the gun crew stay relatively protected and keep firing. If you're familiar with the adoption of helmets in WW1, it's a similar line of logic. The metal helmets adopted by all the powers weren't nearly thick enough to stop rifle fire - they'd be too heavy if that was the case. But they were enough to protect a man's head from flying shrapnel, and they were extremely effective in that respect.