I wondered why; a quick (and lazy) Google scholar search suggested some answers: dog whelks, crabs and oystercatchers all eat mussels and there are plenty of all three on Runswick beach. Shore crabs (of which there are plenty in Runswick rockpools) are described as 'voracious' and 'rapacious pests' when it comes to mussels in one paper (The mechanics of predation by the shore crab Carcinus maenas on the edible mussel Mytilus edulis, R. W. Elner, Oecologia, 36, 333-344, 1978). Dog whelks go for young, thin-shelled mussels and flocks of oystercatchers patrol the receding tide.
Starfish also like wrenching open and eating mussels, although I've yet to properly look for starfish we have found a few dead ones. So mussel predators are present, as is a strong population of other shellfish competing for space on the available rock.All these may account for the absence of Runswick Bay mussels.
In summer, the field assistant and I will take a low-tide walk around Kettleness headland to see if there are any mussel beds to be seen there. Here's the inside of the shell covered in nacre, a form of calcium carbonate. The blue blob at the bottom right of the shell is the scar where the posterior adductor muscle which opens and closes the two shells attaches.
* It turns out byssus threads, which are secreted by the mussel's foot, have more uses than simply gluing the beast to its chosen home. As a free swimming larva the young mussel throws out a byssus thread which acts to slow the larva's rate of sinking through the water. In established mussel colonies, byssus threads are also defence mechanisms; a rash marauding dog whelk might get snagged and immobilized by a byssus thread and, unable to maraud any further, starves to death.