Ummmm - this is supposed to be a supportive forum - I feel even I have been lectured to!I stay out with the Ex Batts. They are not, and their keeping arrangements are not, suitable for free ranging. We hardly have any day time predators, perhaps a fox getting hungry enough to hunt in daylight, or a Kite that just has a go.
In Catalonia I made a decision to free range all the chickens. This meant some were going to get predated. It doesn't matter in the end how you keep chickens eventually for the vast majority of keepers something is going to get the chickens.
Because of the environment in Catalonia which was reasonably good for free ranging including temperatures and forage I was able to do this. I was also able to let the chickens breed and replace the losses from within a closed group arrangement. The losses were terrible before I got there because there was this half hearted semi free range add to tribes through incubation and lots other practices that are common to the standard backyard chicken keeping model.
As the chickens adapted to being fully free range they learn't how to make use of the available cover, how to move from one area to another by the safest routes and many other skills that free range chickens need to learn to survive.
For the first three years most days I was outside with them, plus there were dogs and other farm animals. All these factors helped to reduce the losses while they and I learn't how to deal with freedom.
Your keeping arrangements and many if not most backyard chicken keepers are never going to be able to do this. The backyard chicken keeping model doesn't allow for chicken freedom. Most backyards don't even have enough space to allow for this and it seems many do not allow for roosters so the celf replicating group isn't possible either.
People who have dogs for example should accept that for a period of time each day they need to devote that time to walkiing the dog. Responsible dog keepers who have active intelligent dogs may spend two or three hours a day taking thier dogs out if they do not have sufficient acreage for the dog to wander. My daughter here has a Husky. He's old now but he is still a two hour daily commitment. Chickens, if you don't want to keep them prisoners in the coops and runs are the same.
So there is part of my answer. You can get your tribe out of their run every day if you are prepared to be out there with them concentrating on the chickens and not trying to do other things as well.
Some people go about the problem by actively eradicating local predators. Most do not post about how they "got rid of" some hawk or fox that was taking their chickens but they do it.
I've mentioned before that a sling shot using clay balls will keep you out of trouble if some common sense is used. I realise you may not like this option but if one wants to keep chickens one has to deal with the more unpleasant aspects as well as the better bits. I didn't like having to kill the chickens that got hatched that I couldn't house, or the chicks that the mums abandoned at the nest hanging half alive out of their shells, or killing the sick or too badly injured that I may have known for more years. These things have to be done.
The propoganda that chickens are easy to keep only holds true if one is prepared to lock them in cages, deny them their basic instincts to breed and forage and roam along with the many other activities that single sex groups kept contained are never going to learn and experience.
So, get yourself a slingshot. Learn how to use it and "discourage" the hawk.
Put a couple of hours aside every day to be with your hens and supervise their time out of the run. Even prisoners get an hour a day in a yard these days.
If we can all agree to disagree on the many ways to care for our pets and children we shall all get along well.
I applaud your dedication to your chooks, but 99 percent of people are not in a rural setting where they can dedicate their chooks to a feral existence.
Even I, who does live in a rural setting and where there are thousands of farm chickies abound rarely allow the birds total freedom! Bears, lynx, wolves, Fishers, fox, racoons, weasels, skunks, and owls, hawks, eagles, etc are just too, too many to risk a flock. We don't have the length of seasons to 'make' more baby chickies - a breeding season here is about 3 months.
Maybe in Spain it can be a real viability, but here in the great white north, not so! We make do with what we have at hand and we make the best of what we have - some as we all do!
FYI you must submit some pics as tax, I for one would love be to see some of you feathered friends

My tax: