Matt, your invitation for other possibly novel solutions for the “hierarchy problem” invites an idea about why the Higgs field is locked at a relatively small value.
“The solution to the hierarchy problem involves a completely novel mechanism. Easy to say — but got any ideas? Arkani-Hamed gave us two examples of mechanisms which he had studied that he couldn’t make work — but perhaps someone else can do better.”
So here is my half baked idea, not entirely novel, as it is a composite of several older theories jumbled together: Should a large volume in three dimensional space, (make it universal in size to avoid boundary questions) be filled with charge that has the property of mutually repelling all other charge. Generally easier to visualize as point charged particles. Now, although the pressure and volume energy is large the total average field is near zero as long as the point charges are stationary. That is the charge in any direction is balanced by the charge in the opposite direction. However, this would be unstable because the effective shape of stationary point charges is spherical. And point charges in motion relative to all surrounding point charges take on a shape that packs more efficiently than stationary point charges thus reducing the pressure and volume energy. Any change in kinetic energy of the charge elements (increase or decrease) would reduce the packing efficiency and increase the pressure. This kinetic charge structure generates a non zero field. Or you can view it the other way around and say this universal non zero field produces the charge.