You might want to scan this thread, it’ pretty long. But someone let a broody hen raise chicks with the flock in weather a lot colder than yours. You might want to see how they did it. It is harder for a broody hen to raise the chicks in the winter, what might be a little inconvenience in the summer can be fatal in the winter, but as you can see, it can work. It can be harder for you too, say you get a power outage.
https://www.backyardchickens.com/t/947046/broody-in-michigan-winter
There is no one set way we do any of this. We are all over the board. Some let the hen hatch with the flock and raise the chicks with the flock. Some isolate when incubating or hatching. Some isolate the hen and chicks for different lengths of time, some take the chicks away and raise the chicks themselves. While I don’t isolate them I don’t consider others wrong, just different to me.
If you let the hen raise the chicks with the flock, she will handle integration for you. The chicks will still have to handle their own pecking order issues when she weans them, but they will do that by forming a sub-flock and avoiding the older chickens during the day until they mature enough to force their way into the pecking order. Brooder raised chicks do the same thing, but you have to handle integration instead of the hen.
Most broody hens do quite well raising their chicks with the flock. I’ve never lost a chick to another adult flock member doing that and I’ve had a lot of broody hens. But I think a big key to that is how much room she has to work with. If space is tight she can have problems, but if space is that tight you will probably have problems when you try to integrate them later.
I don’t know what to tell you. I don’t know your facilities or set-up and I don’t know you. Since you haven’t had a broody in good weather before I can certainly understand your nervousness, you haven’t seen how well that hen can cope. If it were my flock with my set-up I’d let the broody handle it all, but I don’t know if that is the right answer for you.
There is a consideration that makes me nervous though. She has nine chicks. They are going to grow really fast. In a few weeks she will not be able to cover all of them. Those chicks can handle cold a lot better than most people would expect, but I’d be a lot happier if you only had six or seven. If they grow up outside they will feather out really fast. I’d think that at four weeks they’d be OK with the broody, just occasionally needing a bit of warmth. But I’d have concerns in the three to three and a half week time frame. You might watch them and see what the temperatures are when they get to that point and see how well they are covered at night.