In The Prisoner of Azkaban, Hermione forgets to go to Charms after going to whatever other class she has that day--Muggle Studies, Runes, or maybe Arithmancy. Whatever class it is that is always the same time as Charms, can it really be true that Hermione is the only student in both of these classes? So shouldn't other people, like Ernie Macmillan for instance, also need Time Turners?
There's no evidence that anybody else had a Time Turner and really no reason why they should have. Timetabling is a dark and difficult art, way beyond me, but it can be done. For example, at my old secondary school, we used to have a two week system, where the timetable would alternate between weeks, so the first week of term you'd follow timetable A, then on the second B, then on the third A again, and so on and so forth.
Nobody else but Hermione was taking everything in year 3, they took just ten subjects. Timetabling ten different classes with no clashes isn't actually very hard, it's pretty standard to do 10 GCSEs in the UK and nobody in UK secondary schools needs magic to fit in all their classes.
But if everybody else is taking ten subjects and the brilliant student is the only one wanting to take twelve, that could really throw your timetable out. Now other people might be told to drop one or two (maybe some of them were) but with Hermione, special arrangements seem to have been made.
The houses also adds something. So, for example, Care of Magical Creatures classes were shared with Slytherin and Gryffindor students both doing the class at the same time, whereas in Divination it appears the Gryffindors take the class only with other Gryffindor students. So that can both give you options (oh we'll put the Ravenclaws in with the Hufflepuffs for Care of Magical Creatures at 09:00) and make things difficult (oh but that Ravenclaw wants to be in Muggle Studies and we've got Muggle Studies at 09:00 and the Hufflepuffs are having Care of Magical Creatures at 09:00).
Timetabling is a dark art, and it's basically impossible to speculate on how the balance worked out in Harry's third year, but there's really no reason to be surprised that everybody else managed to make 10 classes without magic. There's also little reason to be surprised that that one difficult student wanting to do all classes, unlike anybody else this year, required special arrangements.
I'm sure there were other gifted students running around with time turners. But apart from Hermione, the people Harry hangs around tend not to make school a priority. WIzards are used to keeping major secrets, after all. If Cedric Diggory had a time turner and didn't want you to know, would the world look any different from Harry's POV?
As @CBredlow has nicely put in the comments section, Professor McGonagall (not sure with Dumbledore, though) has to go through a lot of paperwork before convincing the Ministry to let Hermione have one, cause she is an exceptional student. She was taking all the subjects offered in the semester, the highest by a student in that year (She dropped off divination, of course).
By the tone in which Dumbledore wars Hermione and Harry about the device and usage, it is quite evident that Hermione might have had some really long sessions before the headmaster and the professor actually gave her the device.
"You must not be seen" is the first rule for the device usage, and Hermione also explains what terrible things happened to people who have not taken that advice seriously. People actually got killed. It's just too difficult for a single person to manage. So, just imagine how hard it'd be to upkeep the rule if two people have it. And remember: Harry saw himself in the Dementors scene. Luckily, it hasn't led to anything nasty/dangerous as he didn't completely recognize himself from the white patronus smoke.
So, as a headmaster of an institute, Dumbledore cannot afford such high risk, and I'm pretty sure he and McGonagall went through some really intense discussions before even considering a filing in the Ministry for Hermione.