I'll add my opinions to this.
We have a flock of 26 hens. The coop is about 10x12 with a large run attached. So far headwear cloth is working and we’ve never had a predator inside the coop or run. I have lost a few to predators while they were free ranging. Fox, coyote, hawk and dogs are our main predators.
Fox are typically ambush predators. They can strike before the rooster even knows they are there. I'm not sure how coyotes hunt. A rooster can't really stand up to a fox, coyote, or a decent sized dog. Oh, anything can happen, it's possible a rooster could interfere if he really attacked, some predators may run away if they feel threatened or surprised. But typically the rooster would die. I've been through two different dog attacks, lost 8 chickens in one and 5 in the other. Both times the rooster did not lose a feather. He obviously tried to lead his flock to safety instead of fighting a rear guard action.
Typically hawks do not carry the chicken away unless they are feeding babies in a nest. Usually they eat where they kill. A rooster could chase a hawk away, maybe even in time to save the chicken. But often they don't. Hawks have delicate bones that break easily, they don't like a fight.
A good rooster keeps a watch out for predators, especially in the sky. They tend to have an instinctive worry about flying predators, more than ground predators. So a rooster may be an early warning system, but if a flock doesn't have a dominant rooster the dominant hen often takes over this responsibility.
When the flock detects that something just isn't quite right a rooster will usually place himself between the suspicious area and the flock. The entire flock gives a certain alert sound, sort of a nervous purring. He very carefully checks it out to see if he needs to lead the flock to safety. That makes him more vulnerable and could contribute to the stories where he bravely sacrificed himself to save the flock. The dominant hen might do this if there is no dominant rooster in the flock.
I’ve also had an issue with a few of the hens ranging beyond our land and tempting the neighbor’s dog.
This describes another issue. With a flock that large the hens are not going to stay in a tight flock around a rooster. Different groups of hens are going to do their own thing. If he is not right there he can't protect them. If they are close enough they might hear a flying predator alarm, but in a flock that size some probably won't be close enough.
Since they’re not out all the time would a rooster (or a few) be super helpful or more trouble than they’re worth?
That's for you to decide. It may take some trial and error to get a good feeling for that. For many people having a rooster doesn't cause any trouble at all. For others having one or more roosters can be a huge problem. A lot depends on the roosters. If you try and it doesn't work out you need to be willing to get rid of him or them.
Is my coop big enough for multiple rooster/hen groups?
The size of the coop may not be the big issue, the size of that large run may be more critical. Sometimes multiple roosters can live together in a relatively small area but usually the more room you have the better. It depends a lot on the individual roosters, especially if space is limited. Sometimes they can share a coop at night but can't share space during the day. The only way to know for sure is to try it. Be ready to isolate them from each other or from the flock if the need arises. That need can come pretty quickly.
