Here we have a lot of "one pipers". It is just a pipe that drains to daylight (downhill at 1/4 inch per foot), The pipe is cheap field tile (plastic pipe in 100 foot rolls that is super cheap to buy). A trench is dug and then half filled with rock. Then the pipe is put on top and back filled with
hay and then soil.
It is only a foot
underground, but the water is draining and warm so it does not freeze like you would think, even in our Maine winter. And it would take just as much time to do as it would to build your system.
Your system will not work and be like Tyley Luden says. You are right, your design is a filter, but what you do not understand is, your filter could not handle the sudden influx of water it would take when you took a
shower for instance. The filter you designed is slow acting, where as a house gets periodic moments of heavy flow.