Auto white balance usually works pretty well for me and like Bob above, when it doesn't work I just correct it in Lightroom.
The fact of the matter is that white balance doesn't need to be precise or exact. Close enough is good enough and what's close enough is an interesting question. Our eyes adapt to differences in colour temperature quite well and if you're indoors at night under tungsten lighting you rarely notice that the colours of everything in your room are significantly different to how they look in daylight. An interesting problem is what you do when the thing that attracts you to taking a particular photograph is the way the colours look, like a strong colour cast at sunrise or sunset if you're walking from indoors under artificial lighting to outdoors. You only see the colour cast in the brief few minutes before your eyes and brain adapt so when you take the photo, how do you want it to look. If you use something like a white balance target and set your camera to that, what you'll end up with in your file is an image in which whites look white and the colour cast has disappeared. The colours in the image will not be the ones you saw which attracted you to take the photo. You want the wrong white balance setting, one probably very close to the white balance indoors that you just walked out of, if you want your photo to reflect the way you saw the scene.
And then there are those times when you simply want to change the white balance for effect, not for accuracy.
I've tried shooting the scene first with a white balance target in it and then without the target, using the target in Lightroom to determine the correct white balance, and then adjusting the actual image, the one shot without the target in it, to the right white balance setting and found myself often unable to really pick a difference because I'd shot the scene using auto white balance and while setting the white balance based on the target changed the temperature and tint settings a little, the result was barely noticeable and didn't really affect the look of the image.
One thing I'll warn you about, however, is that if you're going to use a white balance target like the x-rite colour checker, make sure you put it in a part of the scene that's receiving the light you want the image balanced for. Colour temperature will be different in shade or shadow to what it will be under the lights or sunlight. If you're shooting a sunny landscape, put the target in a shady area and use the result you get from that for your sunny landscape scene, it will look decidedly off. Using things like white balance targets or the like and using them badly will have you wondering why anyone would ever bother using anything other than auto white balance.
Basically, auto white balance works pretty well most of the time these days. Shoot RAW and adjust the image in processing if necessary. There are times when trying to be more precise than that is worth the time and effort but it's rare that most of us really need that level of precision. There are professional jobs when it's absolutely essential but the rest of the time close enough really is good enough.
As for why can't the camera get it right, the answer is simple. Unless you have perfectly dead even lighting across the whole scene with no shadow area at all in the scene, there will be variations in colour temperature across the scene. The camera doesn't know what part of the scene is most important to you and it also doesn't know where you were looking immediately before you saw the scene or what colour temperature your eyes were adapted to at the moment you took the photo. There's absolutely no way a camera can nail the colour balance in a scene exactly the way you saw it every time. No matter how smart they try to make the system it will still get it wrong some of the time and most of the time it's not going to be far enough "wrong" to make a difference anyway. Take a look at the accuracy of the camera's light meter results. My camera's have around 5 different light meter modes and I can get pretty much identical results with all of them provided I know how to use them but how often do we find ourselves adjusting the exposure of the image in processing? When we adjust highlights and shadows separately, we're really giving one exposure to the highlights and one to the shadows. Because of the dynamic range of the scene compared to that of the sensor, there's no "accurate" exposure which will give us what we want without us needing to make highlight and shadow adjustments a lot of the time. The camera simply can't get it right all of the time because the sensor is incapable of capturing the full range of the scene. Things are no different with white balance. I shoot with the camera set to auto white balance and I make white balance adjustments a hell of a lot less often than I make exposure adjustments so I really don't think our cameras are doing all that badly with white balance.
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