Why didn't Harry acquire a working Time Turner / make someone else use a Time Turner as soon as he found out Hermione was missing? Why did Dumbledore make sure it was actually Hermione who died, making the use of Time Turners to alter events that much harder?

If this story wasn't constrained by any story telling concerns, Time Turners would just dominate everything. "How do I use my Time Turner to cheat?" would be the first and last thought of a rationally empowered Harry whenever he has an inkling of something bad happening, and not something he could ever miss.

The story says "This [Time Turner] could've saved Hermione, if I'd been able to use it.", but really, how hard would it be to acquire one of the other Time Turners he relied upon for sending the messages? Even if those students weren't present in the Great Hall at the time, as soon as his own Time Turner was locked down in the first place, acquiring access to another one would have been the highest priority by far, and surely within his means (given that other Time Turners are only controlled by fellow students).

The more I think about it, the more an even more rational version of a HP-fanfic should be called "Harry Potter and the Time Turners". Probably less entertaining, but that's the downside when you take your characters' intellect seriously ...

Edit: I also missed this: "if I'd looked for a student with a Time-Turner to send a message back in time before I found out about anything happening to her, instead of ending up with an outcome that can't be altered". That's one remark that is just wholly incompatible with any half-competent Harry. It's just inconceivable, on the order of someone giving you a genie and endless wishes, and you just not thinking of it in a situation of great need.