Well there's a number of factors that come into play, but first and foremost, the reason a camera operator pulls focus (if shooting alone) or having a 1st AC/2nd AC pull focus is because if you care about what you're shooting, you want focus to be tack sharp.
Focus pulling is required on today's full frame and large sensor cameras because the Hyperfocal is extremely tight/small due to the gate (sensor) size, especially so when using fast lenses, shooting at wide apertures, or long focal lengths. The Hyperfocal Distance is in today's digital cinema world, a very specific and critically narrow point in space, or better put, distance from the the camera's gate. Large sensor cameras also typically need faster lenses, due to the fact that the light entering the lens is refracted over larger plane than on a small sensor capture device. (More info below on that).
Many DSLR cameras have AF features, but they are relatively limited, and require the user to set a focus point, such as center, or another XY or pattern of XY points on the screen in which the camera would attempt to lock to.
Professional systems by RED, ARRI, and others, have 3rd party accesories which can assist with Auto Focus, but again, they are only as good as how you program them.
That said:
With a large sensor, shooting on a fast lens with even a moderately wide aperture at anything other than a "wide/medium wide" focal length will produce an image which has such as shallow depth of field that you have to visually adjust the focus based off the precise distance from gate to your desired focal point.
Do you want the eyes and eye lashes to be tack sharp? Or the tip of the nose? If you focus the camera to be tack sharp on the front/tip of the persons nose, the eye lashes and eyes especially, will appear soft, especially at 4K and above.
You can also focus too deep, which is a very easy mistake to make, making the hairline/front of ears tack sharp, again, leaving the eyes and eye lashes, and even more, the nose, soft and out of focus.
Even shooting a simple sit down interview with someone on a HD camera (not 4K), using a 100mm focal length, at f2.8, this becomes highly noticeable. Tip of the nose, ear lobes, or eye lashes... You want to focus on the eyes... because that's what viewers focus on when watching the footage.
Theres no way a camera can "know" exactly what you want it to focus on. Secondly, automatic systems never work perfectly. This happens often even at short focal lengths using wide lenses; where the cameras AF system will latch onto a brighter spot, such as a light bulb or candle flame far in the background behind the subject- so even on wide shots AF doesn't do a great job.
Lastly, if the AF system gets confused or if something moves in the frame, the AF system will fish for a new focus point, often times over and undershooting the focus point in the process. This fishing effect is highly distracting.
So in closing, if you're shooting anything with actors where their performance matters and you may not get a second chance to capture that exact moment or performance, you have to have someone watching and pulling critical focus on a monitor that is designed to do so (showing pixel to pixel or magnification on a specific zone).
Now with cameras that have a very small sensor, such as the old video cameras from the 90s and early 2000s, the sensor was so small (sometimes as small as 1/5"), that A.) The small gate/sensor size created a very deep depth of field/large hyperfocal, B.) The small sensor received a much higher level of light concentration from the lens (described below); and C.) The small sensor cameras in the past typically shot at lower resolutions and focus could be slightly off, and due to the low resolution it wasn't visually noticeable.
When shooting with a small sensor, the light entering the camera was far more concentrated on the small sensor area (think of a magnifying glass with the sun trying to burn a leaf). The smaller you focus the light into a smaller area, the brighter it becomes. If you pull the magnifying glass up the focus of the sun becomes larger, less bright and less hot.
I only mention this because this is why large sensor cameras require more light. Because the image entering in through the lens is being spread out over a larger canvas, and thus is not as concentrated / bright - requiring either more light, or shooting at a wider aperture. Wider apertures = shallower depth of field.
So the small sensor cameras can shoot in AF mode with greater ease- the reason being "everything" is in focus, due to the small gate, and the fact that the small sensor requires a smaller aperture/f-stop. These factors create a hyperfocal which is so large, that everything is reasonably in focus and there's no discernible difference between the tip of the nose and the front of the eyes.
