It seems like it would be a very helpful device.
Why was it never used again?
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
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:
Hermione confirms this in the beginning of HBP, when explaining to Hagrid why the trio didn’t take Care of Magical Creatures:
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.
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.
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.