Contrary to what you may have heard, trawls travel just above the sea floor with the 'ground-rope' just touching the bottom every now and again - they aren't like the big bulldozers or vacuum cleaners some make them out to be.
If/when the ground-rope digs in then that's called 'coming-fast' - which is something you try to avoid doing at all costs because it ruins the gear, rips the belly of the net and is dangerous - especially if the weathers a bit shitty. Not only that but you cant catch fish if the boat isn't moving!
It's dredging which you are thinking of I think love - like oyster/scalloping. They are the true bulldozers which plow the sea floor.
Bottom trawling generally isn't as harmful to the seafloor as most believe because a trawl net is designed to
herd the fish and the net itself is actually very delicate - dragging one along the bottom like you imply (digging fish out of the mud) isn't something which isn't done.. not on purpose anyway.
The Doors (the things that spread the wings of the net) should touch the bottom more than the net - but their foot print is normally no more than a few inch's and they make a lot of noise - anything in their way has time and warning to move before they get there.
In the defence of oyster/scalloping - most of the bed's worked in NZ now re-seeded via funds provided directly by the fishermen, and the areas where oyster/scallops grow best are muddy bottoms where not much else lives....