I don't think the problem is with the Shotgun in most games, it's with level design and running abilities.
In most first person shooters, players can still run very fast for very long periods of time and most levels consist of mostly close quarters with lots of blind turns. If the vast majority of your encounters are going to be at close range, why not chose a weapon that can down an opponent in one shot? It's a specialized weapon for close-range combat, so of course it should excel at it. Shotguns in most games play to players who are fast and play aggressive, looking to get close to enemies and finish them quick. When range is rarely an issue due to level design, power and spread become the selling points, and shotguns win.
The Clark shotgun from Battlefield 2142 was a surprisingly difficult weapon to score well with, but I thought it was how Shotguns should be done in games. It was a fairly good weapon, but was probably more what I thought a shotgun should be; powerful but unreliable. Turn a corner with the Clark, one shot down a guy with a shot to the chest. Turn another corner and see a sniper lying down, aim right for the upper chest/head, first, and the player turns around angry with their knife looking for payback. Compared to the virtually fail-safe Voss (upgraded Assault Rifle, at fully automatic was awesome at close range, and single shot great at medium to long range) using the Clark was a gamble. But if you played smart, stuck to the areas of the level that played to the Clark's advantage it could be a very useful weapon, especially when playing very aggressive.
The original Halo I think is a good example of how a Shotgun can be wrong. Once you got the Shotgun in the singleplayer, it outclassed basically all of your other weapons. No need to use plasma weapons to down shields then conventional to kill an enemy, you could just run up to them with the shotgun and half the time the first blast would do the job. If for whatever reason it didn't you could follow up with a second blast fast enough to make it a non-issue.
But in the end, I think most shotguns aren't broken, it's levels. Most first person shooters still rely on a lot of corridor running close range that are one step up from the Maze Screensaver on Windows machines. Balanced levels with open expanses and close range areas that allow all weapons to shine is the way to make shotguns feel less cheap or annoying, I think.
And I'm pretty sure the "I like to keep this handy for close encounters" line was originally from the film Aliens (when the ammo is collected from the Marines, one of them takes a shotgun off his back and says the line before they continue onward). I think Call of Duty 4 was paying homage, or else it was a bizarre coincidence considering the circumstances.