Well I can understand how you feel about the TT since I had many doubts about it myself. But after mulling over it a lot, I think it's quite acceptable the way JKR put it (remember it's a magical world after all😉). See the thing is, the Time-Turner doesn't really change anything literally, but gives you a chance to work on something, something that you wish to work on but think there's not enough time for----provided the deadline is not missed.
This can be explained with regard to both the scenarios mentioned in the book:
1. Hermione using it for classes/studying
2. Sirius' rescue mission
In both cases Hermione is aware of how to use it and of the pros and cons of being spotted by her former self or others. The deadline in situation 1 is, for eg: the time of the class she has to attend after the one she has attended by using the TT. And in situation 2 it is before Dumbledore locks them in the Hospital Wing.
So coming to your question, the TT does not rectify anything but like I mentioned in the riddle too, it provides an opportunity to rectify-----before it's too late (In the case of PoA, before Sirius can be subjected to the Dementor's Kiss). "Nothing can be changed because anything a traveller does would merely produce the circumstances he had noted before travelling."-Theory of Time Travel. 😊 Examples that can explain this is that in the actual time, Harry was also hit by a snail shell, and Hermione still heard the crack of a branch - which was what happened in the past.
The only fault, well not fault but deviation rather, that I found was that, in the book when the Time-Turner is used it takes the person back to the location where they were present at the time they'd gone back to. However, in the film, when the Time-Turner is used it leaves the person in the same place they were when they turned time back. 😆
Phewww...sorry for the long post, but concepts like these, take a lot of attention to detail! 😉 😳
