I have 23x15.25x2 board in cherry. It weighs 15.5lb. I have no problem putting it into a sink (narrow end), clean it with a soap and wash it under a hot water, but agree this is a not a light board to lug around the kitchen. Board without feet might skid on the counter top and you don't want to drop endgrain board on the floor. It might crack along seams.
I don't understand the need to use both sides of the board, and besides, those finger grooves that you see on some board, they are not very deep so I am not sure how effective they are. Finger recesses make more sense, but those make the one-sided board. Also, even with proper care, the board might move and cap, and rubber feet absorb those changes. Without feet, you might get your board rocking.
If a board is seasoned properly by the maker, you can wash it (as I do now), even under running water.