We solved the whole thing even more simply - semi-detached house with a basement and converted attic (so 3 floors + basement), in the basement under the staircase opening there is a tub and everyone just throws their laundry down through the stairwell. The kids even make a competition out of it - whoever throws more parts beside it has to go down and pick them up. We adults can of course aim perfectly.
Washing and drying is done in the basement - the machines can even run at night, through the closed door we hear nothing, although we are very sensitive. There is also an electric dehumidifier in the room, so hanging laundry always dries in < 12h. We had a dryer on the sleeping level long enough - never again!
So I only have to carry up the finished laundry. And no, sand and dirt do not get lost on the way when throwing, if someone comes in dirty, they take their pants off immediately on the ground floor and put on something clean before going to the couch.