For thousands of years people on farms have kept a flock the way it sounds like you do. The chickens are expected to feed themselves by foraging in the good weather months. Where you are every month is a good weather month. The quality of the forage has a big part to play in how well they manage to feed themselves. If you try foraging them on a manicured lawn with no diversity of vegetation and critters it won't work well. It sounds like you have pretty good diversity.
Foraging for themselves they will not grow as big as they would if you carefully fed them a rich diet. The eggs they lay won't be as big as they would be if you fed them a high protein diet. That doesn't mean they are not healthy, just that they are not as big or lay eggs that are that big. It's pretty much how they lived before they were domesticated, except you supplement the food some and probably provide a deterrent to predators.
This thing about providing certain amino acids and certain supplements is a fairy modern thing, probably not much more than 100 years old or such. Those supplements do help productivity and are mainly pointed at chickens that don't eat a lot of creepy crawlies and such. If you are providing all they eat then those supplements in the feed are important, but for thousands of years they got what they needed from forage.
Cinnamon Queen's are a cross between two dual purpose breeds, a Rhode Island Red rooster over a Silver Laced Wyandotte hen if I remember correctly. They are a red sex link but are not the kind of hybrid red sex link bred to be the high volume commercial egg layers. I'd expect them to do as well free ranging as any other dual purpose breed.