If you are on well water & the water is OK for fish coming right out of the pipes (doesn't need to equilibrate in terms of dissolved gases) this might very well work. If not you could probably set up an automated reservoir that would let the water sit for gas exchange or do dechlor for a day before it was added slowly to the tank (apparently running an airstone during this process speeds things along). So the water would fill up the reservoir, sit for 24 hours & then be released into the tank only to have the reservoir be refilled. Not quite total flowthrough but close. Chlorinated water or water with chloramines will definitely be harmful to your fish & your snails.
If it where me I would put concrete blocks in the wall under the tank to make sure you won't have any collapses. I would also build in an access to the back of the tank in your utility room. Apparently the problem with most in wall tanks is that they are too narrow to keep fish happy & have insufficient surface area for gas exchange. So the best scenario would be to use a normal tank that could stick out a bit into the room behind the wall so your fish would be happy. If you do go with the narrow tank keep it long & shallow & just get small fish that are happy with a small turn around space.
I personally am also a fan of planted tanks because plants take up the wastes/nutreints that the fish put out from digesting fish food. Plants are also found in the natural habitats of almost all fish & tend to make them more relaxed & less aggressive. There are some plants that can live with chichlids I believe & they would also compete with any kind of algae that decided to form. However, I wouldn't run a tank without a bacterial filter as well (at least as a safety net). I would just be afraid of a plumbing problem & fish overwhelmed with waste. Even a small HOB
filter would make a difference. Dead snail & wammo ammonia spike. Yes your flow through would carry out the waste eventually but why not have a few plants (anacharis even) which would damp those spikes down & not stress out your fish.
In terms of not cleaning, the flow through system could certainly simplify that if it where properly designed but would make little difference if it weren't. In order to get the mulm that settles at the bottom your overflow would need to feed from the bottom & since tanks have corners & usually decorations there are always places that it can get trapped. A rotating powerhead if they build such a thing would certainly minimize cleanout. I would move it to a new location like once a week to maximize its effectiveness. You will also have to clean out the filter/screen out the inflow tube that leads to the overflow. You might also want to design the tank so that the inflow to the overflow is slightly lower than the rest of the tank (the accumulation of debris is usually in lower areas so this could help your not cleaning idea). Maybe to protect it from the digging chichlids you could put it in a loose bunch of rocks. You will have to do some maintenance though so you might consider some algae eaters to help you out. Snails actually put out a lot of waste even though they do eat algae. It might work great with a flowthrough, I have only ever tried this with tadpoles.
This sounds like a really cool idea but it might be a little less simple than you originally envisioned in terms of water flow. I think the turnover rate of water will depend a lot on your bioload. So you probably want to figure that out ahead of time with a little margin incase you want to add something. Keep us posted with new crazy ideas (I love them).
P.S. Hopefully you have plenty of water in Minnesota . . . You could plumb the outflow to go into a flowerbed (fish tank water is supposed to be great fertilizer).