Like many others here, I don't use beginner weapons
exclusively, but I usually use whatever weapon I find best for the situation, which may very well be the starter weapon. In the Half-Life games, that holy instrument of divine vengeance the Crowbar is a very useful secondary weapon, which can deal a lot of damage very quickly, and ofc it's just fun to clobber headcrabs to pieces. The Pistol, at least in HL2 (I haven't really played HL1 much), is also a reliable, versatile weapon with plenty of ammo available, it can achieve a very respectable firing rate and four shots (which can be done in about a second or so) to the crab will kill both a zombie and his critter dead (or re-dead as the case may be), while, for instance, using the SMG may lead to only about two dead zombies (if that) per clip (yeah, I probably suck), and since SMG ammo is rarer than pistol ammo, it's usually the pistol for me when facing zombies (if I don't have any shotgun shells left, that is).
The first Halo would be a similar, well-known scenario. The Pistol's scope makes it the perfect tool for headshotting Grunts at range, saving the ammo of more precious weapons for Elites and other powerful critters. Also, the Assault Rifle, which is admittedly arguably not the starting weapon, is still a good backup in many situations - it has a pretty good damage output against anything, and only the Shotgun beats it in close combat (in the first game, singleplayer, at least).
In Left 4 Dead, I don't use starting weapons much after I find the first stash of second-tier guns, except ofc when I'm down (which happens a lot) and I'm forced to use the pistol, and I rarely really run out of ammo. The one exception was when I was trying to get Akimbo Assassin (which eventually succeeded, after several games that I ended because I got bored) in singleplayer.
In Red Orchestra, the gun I'm using depends on what classes are still open, but I'm not averse to just using the bog-standard bolt-action rifle, those things are still amazingly useful. Although I do go for the semi-auto rifle or light machine gun whenever I get the chance. I rarely take a class with an SMG (unless it's the tank commander class), and I'm not good enough to snipe.
Iron Mal said:
In most other games (with, as someone said, the possible exception of RPGs) your starting weapon is often a balanced and versitile weapon suited for most situations.
Newer players can also have trouble figuring out what gun to use at what range. While playing Halo with my sister she would often ask me why her shotgun rounds weren't hitting anything (my response usually being 'you're too far away'), as you can see, it's much simpler to have newer players use one weapon all the time rather than constantly assess the situatuion and switch between three.
That was one of my love-hate things in Halo 2. On the one hand, you have to really consider your loadout, on the other the lack of all-rounders can be frustrating when you're faced with an unexpected situation, and nabbing an appropriate gun off of a corpse isn't always an option, and even so you might not want to lose your current weapons. (Especially if they include such juicy things like the Rocket Launcher or one of the sniper rifles, which can be VERY useful.)