I had a very severe head impact couple of years back, and while I was fuzzy at the time of impact, few hours before I go to bed, it was not until the day after when, my internal functions went half way south. I am not certain if the weakening of some of my external senses immediately happened or not.
This was dead, but it seems like a sensible question so I vouched for the comment to resurrect it.
My understanding is that with any head trauma doctors are concerned about the possibility of bleeding into the brain, and it's much easier to detect the neurological symptoms of this in a patient who is awake. But I'm not a medical doctor; someone else here may be able to provide a more in depth answer.
You're correct; as far as the patient is concerned, it's better for them if you let them sleep. But it's easier for everyone else if the patient isn't allowed to sleep, as sleeping and dying look exactly the same.
I do not know how many dead people you've attended to, but the ones that I have seen generally lack pulse or breathing. Both of those vitals are monitored for inpatients. And if one of those goes, the other goes too in short order.
Sleeping patients, on the other hand, usually pulse at least once per second, and breathe every six seconds or so.
Choking? Pulse goes up and breathing becomes shallow.
Cardiac arrest? Aneurysm? Torn blood vessel? Shot in the head? Stabbed in the chest? Poisoned? Spider/Snake bite? Fell off bed and broke hip? All these "dyings" are easily detected by pulse and breathing monitors.
"and it's much easier to detect the neurological symptoms of this in a patient who is awake"
In this context, I imagine neurological symptoms would be things like cognitive function, spatial coordination, memory functions, and linguistic functions.
All of those are things that are not really possible to assess while sleeping, but would be possible to assess in a patient who is awake.
If the patient is sleeping, you can't use a) or b). Now, there might be an argument that everyone should get c) and lots of sleep, but drilling into a person's head is not risk-free either.