I still have a 3 bin compost system for other compost. That's where the poop from the coop goes, and most kitchen scraps. We have a compost bucket on the counter, and it gets everything, including stuff I wouldn't give the chickens, like coffee grounds and moldy vegetable bits.
I dump all the chicken poo and coop litter right into the chicken run. It all gets mixed in with the organics. But I use the deep bedding method and only clean out the coop twice a year.
I have separate pallet compost bins for moldy stuff and things I know the chickens should not eat.
I can understand not having enough room on the counter for both chicken scraps and other compost waste. In my house, we have one ice cream pail on the counter for chicken scraps and that gets almost all our kitchen waste and leftovers. If we have something that I cannot feed to the chickens, but is good for the compost bins, then we will just put that in a separate bowl to take outside - like in a few hours because we don't have room on the counter to have buckets for everything, either.
I do the Berkeley hot rot method for composting in the 3 bin pile. The pile is pretty big right now, and it's still several weeks before I'll go turn it. Yeah, it's a lot of labor. I do get a lot of compost out of it, so I count it as worthwhile.
I have a 5-bin pallet compost system that I built years ago. I used to turn the piles every once in a while, but never got into all the work required for hot composting. Good for you if you can. It's a lot of work turning those piles.
I let the chickens do all the composting for me. Like I said, it might take 3-4 months to breakdown in place in the chicken run, but I have so much composted material in there finished and ready to harvest, that it is never an issue for me.
For me, that's the real trick to composting. Just make more than you can use and bank the rest in place until you need it. It only seems to get better with age, anyway.
But, @gtaus, I'm a teensy bit jealous of your chicken run compost! Yours looks better than mine!

It's a great system where the chickens do most of the work for me. But I could not be happier with the results of the chicken run compost system I made after reading many threads here on BYC forums and watching numerous YouTube videos on the subject as well.
I really got into mass amounts of composting and built a cement mixer compost sifter which I am sure you have seen, but here it is again for others who might not know about it. It cost me just over $200 to make it with brand new cement mixer about 3 years ago. But it was one of those investments that paid for itself in about 4 hours. I calculated that I can sift out about $60 worth of finished compost per hour by myself, with minimal effort.