Mixing [deep litter in the coop] is just to much work for my family.
I think I should define some terms as I know them. Deep litter is a moist composting system in the coop. If well balanced, it should not smell.
I use a dry deep bedding system in my coop. The idea is the deep bedding will absorb the chicken poo and dry it out. The chicken poo works its way down deep into the bedding, so the coop should not smell if in balance. I clean out my deep litter twice a year and start over with fresh litter.
I don't mix my coop litter. If I see a patch that could use a little work, I might toss some chicken scratch on that area and the chickens will scratch and peck in that area, mixing it themselves.
I don't have enough paper in order to have enough shredded paper for my coop
You don't need to put down all the paper shreds litter at the start. In my case, I put maybe 3 or 4 inches of paper shreds into the coop and add additional layers over the six months until it reaches maybe 10 inches deep. Then I dump the old coop litter into the chicken run for composting. That system works great for me, especially in the winter, because the new paper shreds will cover the frozen poo under the roosts and keep everything looking good.
[Muck buckets are] a good idea...
I like the idea of using muck buckets. But I bought a bunch of cheap $10.00 plastic muck buckets from
WalMart and they are not holding up very well. I only fill them halfway now because otherwise the plastic would crack and break and/or the rope handles would pull through.
Menards sells a similar tub...
I would recommend comparing them to the more expensive muck buckets sold at the farm stores. It might be worth paying more money for a higher quality tub up front and saving money in the long run.

On the other hand, a full muck bucket of old coop litter would too heavy for some people, so maybe the cheaper tubs filled halfway would be good enough. I can lift a full tub of paper shreds litter, but when I used wood chips as coop litter, that was way too heavy for me to fill the entire tub.

How can I [convert my chicken run into a composting system]?
It was an evolution for me. I started out with a nice green grass chicken run that look absolutely beautiful. Within a few months, the chickens had ripped up all the grass and everything was basically bare dirt. That turned into mud after a rain.
So, I first dumped a few loads of wood chips into the chicken run. I can get all the free wood chips I could ever use just by going out to our local county landfill. Wood chips are great and I think I could have been happy just having a chicken run full of wood chips.
But then I started reading about making compost in the run and using that compost to feed your gardens. That sounded like a better system for me. I started dumping all my grass clippings, leaves, weeds pulled from the garden, and just about anything else that was organic that would compost. All my old coop litter gets dumped into the chicken run.
Where I live, I harvest my finished black gold compost about every six months. By volume, it's mostly leaf mold and grass clippings, but of course all the chicken poo from the coop litter is also mixed in. It is some really good stuff. I have way more compost in my run than I can use, but compost only seems to get better as it ages, so I am OK with taking out what I need and leaving the rest to continue composting.
I harvest hundreds of dollars' worth of compost from my chickens run about twice a year. So much, in fact, that I invested in a cement mixer compost sifter setup...

For many years, I just used a manual sifting frame with hardware cloth on top of my wheelbarrow to sift out compost. That works fine, but it is a labor-intensive job. With my cement mixer compost sifter, I can process enough compost in 15 minutes that would take me maybe 3 or 4 hours by hand.
When I first bought my cement mixer and made it into a compost sifter about 3 years ago, I estimated that I was processing about $60.00 of store-bought compost equivalent every hour. And that is at my old man pace not working very hard or very fast!
If you have a backyard flock of chickens, and you love to garden, you are really missing out on a lot of compost potential if you don't put your chickens to work in the run making that compost. And it's not forced labor, either. The chickens love to scratch and peck in the chicken run compost litter all day looking for tasty bugs and juicy worms to eat. All that action breaks down the litter fast. And my chicken feed bill is about half in the summer when the girls are outside digging in the compost and finding stuff to eat.
I should also mention that I live on a lake and have hawks and Bald Eagles overhead all the time. I cannot let my chickens free range without a high risk of losing them to predation. So, I brought the free range to them by converting my chicken run into a composting system. It works well for me and the chickens.