"I'll get yer an owl. All the kids want owls, they're dead useful, carry yer mail an' everythin'." - Hagrid to harry Potter, HP1.
So, what other "useful" things are owls for, aside from carrying mail?
I can't think of any specific things mentioned within the series about owls. However, we do know Rowling pulls a lot of her magical ideas and creatures from already existing lore. If we were to assume she would attribute at least some already existing owl lore to these amazing birds than they really are quite useful in a host of ways.
First of all, witches are often associated with a "familiar." Familiars are a bit like witch pets, but with added benefits. How the relationship works is quite mysterious, but they act very much like an animal version of an assistant in much of the lore. They might spy for their master, gather materials, act as watch and guard over a home . . . A familiar can reportedly be any animal, but cats and corvids (birds in the blackbird family) are most commonly used in this way in the lore I know. Whatever animal is used, its skills and usefulness to the witch or wizard it serves is often related to what type of animal it actually is. Since owls are associate in various cultures with death, message delivery, healing and warding off evil, and finally, with prophecy I'd suspect that as familiars their skills would related to these realms in various ways (depending on the specific owl) - but more on that further down.
I'd assume in the world of HP, it is cats, toads and owls the school is willing to host and finds most useful (for whatever reason) for beginning witches and wizards. Of course, for some reason Ron is allowed his Rat even though this animal is not specified as allowed for first years on the school supplies list.
We see an example of this kind of use of animals in PoA when we see Crookshanks, Hermione's cat is able to distinguish between Animagus and full animal. Crookshanks works with Harry's godfather and is especially obsessed with catching one very particular rat throughout - he never manages to communicate his knowledge to Hermione, but she is a beginner witch so maybe, Hermione hasn't come across how to better listen to her familiar yet.
We do see an example of Hedwig fulfilling (or at least trying to fulfill) an assigned task when Harry tells Hedwig to keep pecking at his friends until they send a proper return answer to him in OoP when Harry is pissed because no one is telling him anything while he is still stuck in Little Whinning. Hedwig does follow through and when Harry is reunited with Ron and Hermione at #12, both kids have wounds from the bird's peckings. While this is related to postal delivery, there is definitely more there than just picking up and delivering packages. Hedwig has to judge Ron and Hermione's responses to determine they aren't good enough to bring back to Harry.
Hedwig is also able to find where Sirius is hiding even though no one in the Ministry of Magic can find him in book four - that is pretty darn useful. And Hedwig does save Harry's life - at least once (if you don't know what I'm talking about, you'll find out in the final book).
Nagini, Mrs. Norris and Fawkes are more wonderful examples of the familiar - they are certainly more than pets and have some sort of intelligence that allows them to understand their master's needs in ways beyond the average animal's.
To get more specific about owls' advantages, we have to go further outside of the books
Real-life qualities an owl possesses that may be useful to someone include, near absolute silence in flight, amazing eyesight, strong talons and an almost 360 degree ability to view the world only with moving their necks. They also hear quite well. It seems if you need a guard (are worried about dormitory pranks etc) having an owl around might well be of great advantage.
If you need bones of rodents for your potions, owls automatically collect a plethora of these, as they eat the rodents, digest what they can and then regurgitate the remains that cannot be digested. That leaves you with an easily dissected pellet of fur and bones.
The lore associates owls with all kinds of magical things depending on the culture in which you take a look. Greeks saw the owl as a companion to Athena so it was associated with wisdom (we still have this association today in many fairy tales and a mock version of it even in Winnie the Pooh). I think the wisdom shown by many of the owls (except Errol) in the books hearkens back to this traditional association. However, Rowling does give us the counter of Errol and shows us degrees in owl talent - Hedwig is definitely wiser than many of the owls we meet. A student preparing essays, and practicing for exams might appreciate having some extra wisdom around, it that wisdom was accessible. But really, even if they can't communicate it or share their wisdom, just having a servant that is wise (as opposed to the opposite) seems nice anyway.
Some cultures believed owl parts can do different things. For example, in Brittany, boiling an owl's eggs till there was nothing left but ash, produced a magical ashen substance that, when used in a potion correctly, could improve eyesight (great for late-night study benders). In Zuni legend, babies with an owl feather in their crib would not be visited upon by evil spirits and if one hung an owl feather over the doorway into the home illness would not enter the house (Only Hermione would appreciate this one, as most of the kids liked sick days as it meant missing classes - especially if it was a potions day). Seriously though, as far as Rowling's choices go, St. Hedwig is associated with Healing and there is a spring that bears her name in Austria. Perhaps, a white owl by this name in particular, would help Harry in coping with his past by warding off some of the anger that could have over-taken him if Harry had let it (No canon - just my own romanticism which is why I use "perhaps").
In North America, many cultures associate the owl with prophecy and vision - particularly as it pertains to death (hmm. . . prophecy and death? In The Harry Potter Series?). In fact, many cultures see the bird as a god of death or at least associated with death around the world. In some, the birds even have the ability to help in guarding against death while in others the birds serve as a warning that death is coming. In fact, in Romania the Snowy Owl (Hedwig) was seen as the spirit of a repentant soul flying toward heaven. I don't have any clue how this might be helpful to a student, but Hagrid was the one that said they were useful - who knows, maybe he finds portents of death a useful thing. . .
In any case, while Rowling doesn't delve directly, in the series into the benefits of having an owl (more than what you have already quoted), I think it is safe to say, she figured that at the very least, owls also would help keep the rodents at bay while they deliver the mail (making it easier for the kiddo's to keep their candy pestilence free as well).
In Order of the Phoenix chapter 3, Harry sends letters to Ron, Hermione and Sirius, asking Hedwig to “Keep pecking them till they've written decent-length answers”. Hedwig does so indeed as in chapter 4 we find Ron and Hermione has peck marks from Hedwig. So that's another useful thing about owls.