You answered one question, chicken density. Kind of high but not ridiculously so, others have worse. The poop will build up some. How well does your run drain? If it is in a low spot where water collects you can have issues with any bedding. If it is where water drains away from it you have a lot more options. Putting a solid roof sloped so runoff runs away from the run will help, but rain and snow will blow in from the sides. It will still get wet. If it drains well that's usually not a big deal. Water is your enemy in a run. If it stays wet for an extended time it can become unhealthy and stink. If it stays dry or dries in a reasonable time it should be OK.
My run is bare dirt, but it is up high where it drains well. Plus I have a big area inside electric netting where they spend most of their time during the day and where they drop mist of their poop. Poop doesn't build up in the run very much. Yours will build up a lot more. You can try bare dirt and see how it works, there is nothing wrong with trial and error. We are all unique in our set-ups, climates, and many other things. It might work for you and would be a low maintenance solution.
Sand has two benefits. It drains really fast as long as long as the water has a place to drains to. If the water does not have any place to drain to it will sit under the sand, the poop filters down through the sand and stinks when it stays wet. Some people reduce the poop load in the run by using a scoop to remove clumps of poop. Sand works well for that. Sand can be a great option for some people, for others it just doesn't work.
People use all kinds of things for bedding in the run. Dry leaves, straw, hay, wood shavings, wood chips, practically anything you might use as bedding for a coop or even in a nest. You can find people for any of them that think they are great, others will say they don't work at all. Different people with different conditions and management methods have different results. Some people turn their runs (or coop floors) into compost piles. In addition to any of the above stuff, (except maybe wood chips that take a long time to break down) they might toss in kitchen or garden wastes or grass clippings, green or dried. Generally if it drains well any of them work pretty well. If it doesn't drain very well you could have problems. You may need to dig it out and dispose of it.
We are all unique with different conditions. Some things do work better than others for certain people. I don't know what the right answer for you is. I'd look at how it drains and if you need to address that. Then try something and see how it works.
Good luck!