Alright, so after managing to get a whopping 6 stars in 7 nights I've managed to figure out the exact mechanics of falling stars.
First of all forget the nonsense about full moons and blood moons. Falling stars occur every night roughly between 9:30pm – 12:00am.
Falling stars always spawn in the direction either link or the camera is facing at the moment they spawn. Not 100% sure if its link or the camera but make them both face the direction you want the star to spawn in.
DO NOT skip to 9pm with a campfire. This will prevent the star from spawning hence the 6 out of 7 success rate was due to attempting to start at 9pm. Let time flow naturally for your first falling star and from then on you will have to wait from noon aka 12pm and wait till 9:30pm – 12:00am.
Note: I was near a lit campfire at the dueling peaks but I do not know if these are factors. It may be that its simply easier to see the stars from a high up spot.
Alright so I left my game on pause for 3 hours in position at the dueling peaks and it seems something I did broke the loop and stars stopped spawning. However, I left the area to kill a lynel and returned to the dueling peaks shortly after and naturally let time pass from 7am to the 9:30pm at which point a star has spawned again.
It seems that if the stars stop spawning for any reason the best option is to leave the area and do something else for a minute or two and return to the dueling peaks and allow time to flow naturally. It's also possible that when you arrive at the dueling peaks for your first star fragment you might need to allow time to flow naturally and only after receiving your first star you are able to do noon jumps.