There is no new work that needs to be done on them (I am guessing on this, since they let kids use them).
This actually isn't the case. The Pottermore entry for Time-Turners says:
According to Professor <PERSON>, who has spent his entire career in the Department of Mysteries studying time-magic:
“As our investigations currently stand, the longest period that may be relived without the possibility of serious harm to the traveller or to time itself is around five hours. We have been able to encase single Hour-Reversal Charms, which are unstable and benefit from containment, in small, enchanted hour-glasses that may be worn around a witch or wizard’s neck and revolved according to the number of hours the user wishes to relive. All attempts to travel back further than a few hours have resulted in catastrophic harm to the witch or wizard involved. It was not realised for many years why time travellers over great distances never survived their journeys. All such experiments have been abandoned since 1899, when <PERSON> became trapped, for a period of five days, in the year 1402. [...]
Even the use of the very limited amount of Time-Turners at the Ministry’s disposal is hedged around with hundreds of laws. 194
While not as potentially dangerous as skipping five centuries, the re-use of a single hour can still have dramatic consequences and the Ministry of Magic seeks the strictest guarantees if it permits the use of these rare and powerful objects. It would surprise most of the magical community to know that Time-Turners are generally only used to solve the most trivial problems of time-management and never for greater or more important purposes [...].
Hence, we can assume the following:
* There are wizards at the Ministry studying time, and there is still work ongoing, because <PERSON> says "As our investigations currently stand", implying that there could be progress.
* They are known items in the wizarding world, and they are complete (in as much as, they work).
* Every single one of them is kept in the Ministry because they can only be used for trivial uses. If they were to release them to the public, it would be utter chaos (plus, even at the best of times, the Ministry is shown to like controlling things). They released a single one, for a guaranteed very trivial time-management purpose (getting to various classes), to people that were known and trusted (<PERSON>, under <PERSON>'s supervision, who vouched for <PERSON> in several letters to the Ministry).
* So, even though people know about them, the details, and access to them, should still be kept top-secret - and there is ongoing research. Hence, they are kept with all the other time-related items.
Note: I'm not sure that your assertion that "everyone knows about Time-Turners" is correct. Certainly, as <PERSON> was using them, no one seemed to cotton on to what was happening, even though she did make some mistakes - it's understandable that <PERSON> wouldn't know about them, but it doesn't seem to occur as a possibility to <PERSON>, who is otherwise quite well-versed in your common wizarding items. Although <PERSON> does mention them later on, in HBP, it's not clear that they were something he always knew about, or if he found out about them from <PERSON> after third year. So, at the time of our first encounter with Time-Turners (PoA), they could still be a fairly unknown wizarding item, particularly if the research was recent, which <PERSON> may have known about as a professor and an exceedingly clever witch. 821