I do see what you mean about the two-weapon system being annoying. The thing is, I imagine it's like that for several reasons. For one, you have to keep in mind Halo: CE was designed for the Xbox console. While Timesplitters 3 did solve the multi-weapon issue one way by having you select a bunch of guns with the D-Pad, it was kinda clunky. Perfect Dark did have a version of the weapon wheel but it didn't slow down time. And even if it did, it obviously couldn't slow down time for MP. So, to cut right through all this nonsense, Halo: CE just limited your arsenal to two weapons at any one time (not including grenades) which could be super easily switched to and from with the 'Y' button. This also allowed them to give more quantitys of more powerful weapons to the player since ammo was now a very important consideration and it gives the less powerful weapons much more of a role.
In terms of the PC though, at the end of the day, the multi-weapon system and the two-weapon system both have equal amounts of advantages and disadvantages. Both can be screwed up without careful balancing.
I know that's what they were going for and it was a fairly elegant solution to the problem and added an extra layer of strategy to the multiplayer, since you still had to collect the weapon like in other shooters of it's time but you also had to decide which weapons you were going for since you could only carry 2, so you had to weigh which was worth it based on the strategies you liked and the risk it entailed to actually going to find the weapon, and like I said I don't remember Halo handling it poorly on the single player (Though I don't remember that much of it) but the fact that it became industry standard and most of the other games used it without considering what would be best for that game, like weighing if they should do a full inventory like in older shooters, doing the 2 weapon thing or the other option of having a limitted inventory of 4 weapons each of which could be assigned to a different direction of the D-Pad, which some other games did back then, each has it's advantages and disadvantages but most game devs just copied Halo's system without considering which fit their game better.
Like I mentioned before a good example would be Red Faction Guerilla, considering how much the game focuses on demolitions they should either have decided on expanding your ammo capacity and limitting your inventory but instead they give you both a limitted inventory and very small ammo capacity which limits you to using the reliable weapons because most of the weirder stuff gets very limitted ammo capacity and isn't practical for killing the huge waves of enemies you have to face, which just makes the game more boring because those other weapons are actually fairly fun to use but you never use them, because even if you decide to carry them you get very few shots or end up picking up an enemies weapon instead because you need it, which leaves the hammer for most of your demolitions, which granted it's fun to use but it just limits your options.
I just use that game as an example because it's one I've played recently but there are plenty of games such as Just Cause 2 which suffer from similar problems, in any case it's not so much Halo's fault but the fault of trend-chasers and I find it a shame, in any case I have nothing against Halo but I never got into the Hype and though I've played 1, 2, 3 & Reach and I'm pretty positive I only bothered finishing 1 & 2 and I don't consider either among my favourites, but since You mention Timpesplitters 3 that is very much the style of shooter I prefer and one that I would consider a favourite of mine, would be really happy if they did a PC port of that one, not a remake or HD version with improved textures and lighting just a straight up port would be fine for me, but I don't think that's happening.
Might be worth mentioning that I've played Halo Single player but it never caught my attention to bother with it and most of what I played of the campaign was co-op.