What @magnetman says about the river below the king's stone certainly tallies with my understanding of this part of the river.
WRT the EA being the legal owners of the river-bed, that is correct. They can't charge for boats passing or mooring over the river-bed, but they certainly charge an "accommodation" licence for any permanent structure over or in the river-bed, so I know people who have to pay for mooring posts driven into the river-bed, and also someone who has to pay for part of a wharf built over the water (simply to create a straight line between his neighbours' moorings, one of which steps out by a couple of feet.