It seems like it would be a very helpful device.
Why was it never used again, regardless of the purpose?
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The Ministry regulates their control and usage, to stop people mucking about with them. When the Time-Turner is first introduced, Hermione tells us that
Professor McGonagall made me swear I wouldn’t tell anyone. She had to write all sorts of letters to the Ministry of Magic so I could have one. She had to tell them that I was a model student, and that I’d never, ever use it for anything except my studies.
— Prisoner of Azkaban, chapter 21 (Hermione’s Secret)
From this, we can assume that the majority, if not all, Time Turners were under Ministry license and control. (It’s possible one or two might be elsewhere, cf. unregistered Animagi). They’re obviously a subject of intense research in the Dept. of Mysteries, as we see in OotP. In the battle to escape, the Ministry stock is apparently destroyed:
The baby-headed Death Eater was screaming and banging into things, toppling grandfather clocks and overturning desks, bawling and confused, while the glass cabinet that Harry now suspected had contained Time-Turners continued to fall, shatter, and repair itself on the wall behind them.
— Order of the Phoenix, chapter 35 (Beyond the Veil)
Hermione confirms this in the beginning of HBP, when explaining to Hagrid why the trio didn’t take Care of Magical Creatures:
“Ar, I always knew yeh’d find it hard ter squeeze me inter yeh timetables,” he said gruffly, pouring them more tea. “Even if yeh applied fer Time-Turners—”
“We couldn’t have done,” said Hermione. “We smashed the entire stock of Ministry Time-Turners when we were there in the summer. It was in the Daily Prophet.”
— Half-Blood Prince, chapter 11 (Hermione’s Helping Hand)
So that explains why the Time Turner couldn’t be used again, from a practical standpoint.
What follows is definitely an out-of-universe explanation, but I include it because I think that it’s interesting.
In many ways, once we know about Time Turners and Voldemort is back, we need to write them out of the story. They’re an insanely useful tool to have in a wizarding war: you could pre-empt and warn yourself of any duel or attack, wipe your opponents from existence or look forward to see the result of a plan. Introduce time travel, and everything becomes very complicated. (That's not to say that you couldn't make a good story from that; indeed, something like Doctor Who thrives on that, but it doesn't fit in Harry Potter.)
For example, you could go back in time and multiply yourself in a duel. Even Dumbledore would stand little or no chance if facing ten Voldemorts. Sure, there are laws governing their use, but Voldemort would be unlikely to pay attention to them. (Plus, he controls the Ministry, so he could repeal them if he really wanted.) Once Voldemort obtains a Time Turner, things go downhill fast for the Order. We remedy the potential damage time travel causes by destroying them entirely.
It's possible that any Time Turners not stored at the Ministry were later destroyed separately, for that reason: to stop Voldemort getting his hands on them. And while the Ministry might have been able to create new ones, they may have chosen not to, for the same reason. Or perhaps, behind closed doors, they might have been working on it, even if it never came to fruition.
Time travel provides a very deus ex machina solution, and had to go. It just so happened that Neville was the way that they went.
Finally, there was another Sci-fi SE question about nine months ago about the use of Muggle weapons that's worth reading. The questions are broadly linked. Introducing guns and/or time travel to Harry Potter would have ruined the atmosphere.
I solved the problem to my own satisfaction in stages. Firstly, I had Dumbledore and Hermione emphasise how dangerous it would be to be seen in the past, to remind the reader that there might be unforeseen and dangerous consequences as well as solutions in time travel. Secondly, I had Hermione give back the only Time-Turner ever to enter Hogwarts. Thirdly, I smashed all remaining Time-Turners during the battle in the Department of Mysteries, removing the possibility of reliving even short periods in the future.
All the Time-Turners, which the Ministry of Magic had full control over, were destroyed in the Battle of the Department of Mysteries in Order of the Phoenix.
Harry stuck his head out of the door and looked around cautiously. The baby-headed Death Eater was screaming and banging into things, toppling grandfather clocks and overturning desks, bawling and confused, while the glass-fronted cabinet that Harry now suspected had contained Time-Turners continued to fall, shatter and repair itself on the wall behind them.
Order of the Phoenix - Page 700 - British Hardcover
The other answers do a very good job of explaining the plentiful dangers of using Time Turners for serious things: erasing people from existence, damaging time itself, and so forth.
From Harry Potter and the Cursed Child:
SCORPIUS: Have you heard me, Albus? This is bigger than you and your dad. Professor Croaker’s law — the furthest someone can go back in time without the possibility of serious harm to the traveler or time itself is five hours.
They do make another appearance, though.
In Harry Potter and the Cursed Child, not one but two Time-Turners are integral to the plot.
They were utilized several times, with various effects:
Albus and Scorpius used them to try to right Cedric's death, while the main antagonist, Delphini, tried to save Lord Voldemort.
Because in the 5th book, The Order of Phoenix, in the epic battle that took place in the Ministry, all the time turners were destroyed.
I reason that as the time turners were stored in the department of mysteries, they may well have been the invention of a very intelligent wizard or witch. Making replicas possibly involved knowledge that was elusive to even the greatest wizards or witches of that age, hence a mystery.