I'm very sorry to hear of the out of control situation with your flock...
To really halt breeding, all nest boxes and potential hiding places that can be used as a nest must be removed from the aviary.
In addition to that, you must separate your flock by gender and place all males in one area and all females in a different area. This can be done by either dividing the aviary to block the access between males and females or by moving your females to another aviary not adjacent to the current one shared by all.
How many pairs are currently actively breeding? At this point I gather a potential inbreeding situation can be occurring.
And given the circumstances, isolating the current breeding pairs from the rest of flock would also be a good option. https://talkbudgies.com/budgie-breedi...s-dangers.html
Given the many issues, you would need to have a better access to the pairs and chicks in order to give them the required assistance, especially if there are problems with leg splaying and a potential clutch that may be carrying French moult.
Having your chicks examined by and avian or exotics vet will also be good.
These links may of use:
I have checked your gallery and didn't see any updated photos on your chicks.
: Upon reading your post a second time, I see there is only one pair breeding and in order to avoid more problems, this pair and the chicks should be isolated from the rest of your flock. While not exactly ideal, and for a matter of logistics instead of moving your big flock, you could try to find a way to block your flock from reaching the breeding pair and the chicks in the aviary. Moving the growing family to cage would allow you to monitor them all better, but you should be aware that there are risks in this and you may need to fully support the chicks if the mother starts to show signs of neglect towards them.